Math HL
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4-5 September 2012
Izmir, Turkey
IB Mathematics Higher Level
&
IB Mathematics Standard Level
Workshop Resource Book
Tim Garry
ibmath.tg@gmail.com
IB Math HL & SL workshop
Page 1
** 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
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S
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A
Y
W
E
D
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S
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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
Page 2
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
Page 3
IB Math HL & SL workshop
Page 4
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
Page 8
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
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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
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Math SL
Math HL
Topic 5 - Vectors
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Math HL
Topic 6 - Statistics and Probability
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Math SL
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Math HL
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Math SL
Math HL
Topic 7 - Calculus
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Math SL
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Math HL
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Math HL
credit: Wiley Miller, Universal Press Syndicate
IB Math HL & SL workshop
permission for classroom use only
Page 24
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
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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
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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
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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
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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
IB Math HL & SL workshop
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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.
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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)
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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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IB Math HL & SL workshop
Page 34
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]
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IB Math HL & SL workshop
Page 35
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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IB Math HL & SL workshop
Page 36
5.
[Maximum mark: 7]
Show that
d cos x sin x
2
dx cos x sin x 1 sin 2 x
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IB Math HL & SL workshop
Page 37
6.
[Maximum Mark: 6]
(a)
Find
(b)
Find
6
x
3
0
3
dx
x( x 3)dx
[3 marks]
[3 marks]
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IB Math HL & SL workshop
Page 38
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]
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IB Math HL & SL workshop
Page 39
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)
x3
.
x2
(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
Page 40
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
Page 41
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)
<3 heads implies 0, 1 or 2 heads
1
10
40
51
243 243 243 243
[1M 1A]
[1M 1A]
[2M 1A]
Section B
8.
(a)
1 5 10 10 5 1
[1A]
(b)
(i)
1 5x 10 x 2 10 x 3
(ii)
[1M 1A]
1 5(0.003) 10(0.000009) 10(0.000000027)
[1M 1A]
1 0.15 0.00009 0.00000027
1.01509027
1.0150903 (7dp)
[1M 1A]
(c)
r
3 15
15 r
3 r x 303r
Cr 2
x
the term is independent of x if 30 3r 0 r 10
15
Cr 2x 2
15 r
hence required term is 15C10 2 5 310
9.
(a)
x2
[1A]
(b)
y 1
[2 A]
(c)
y
(d)
x 3
3
2
[3M
2 A]
[2 A]
[2 A]
IB Math HL & SL workshop
Page 44
(e)
dy
5
dx ( x 2) 2
dy
0
dx
[2M 1R]
There is no value of x for which
no turning points
(f)
1
3
2
3
2
[3A]
10.
(a)
(i)
X ~ N (45,10 2 )
Z
X
P( X 40)
40 in standard units is
40 45 1
10
2
P( X 40) P( Z 0.5)
1 (0.5)
1 0.6915
0.3085
(ii)
[3M 1A]
10000 0.3085 3085
IB Math HL & SL workshop
1M 1A
Page 45
(b)
X ~ N (45, 2 )
P( X 50) 0.0062
50 45
P Z
0.0062
1 (a) 0.0062
(a) 0.9938
a 2.5
50 45
2.5
2
11
(a)
3x
2
1
2
[4 M 1A]
1 dx
2
x3 x 1
10 2
8u 2
(b)
[2 M 1A]
3x 1 dx
9 x 6 x
2
2
2
1
2
1
4
2
1 dx
2
9x5 6x3
x
3
5
1
9
9
(32) 2(8) 2 2 1
5
5
354
[5M 1A]
5
IB Math HL & SL workshop
Page 46
… MOCK EXAM ……… MOCK EXAM ……… MOCK EXAM …
MATHEMATICS
HIGHER LEVEL
PAPER 1
Name
Friday 11 March 2011
Candidate session number
0
2 hours
0
INSTRUCTIONS TO CANDIDATES
Write your name and session number in the boxes above.
Do not open this examination paper until instructed to do so.
You are not permitted access to any calculator for this paper.
Section A: answer all of Section A in the spaces provided.
Section B: answer all of Section B on the answer sheets provided. Write your session number
on each answer sheet, and attach them to this examination paper and your cover
sheet using the tag provided.
At the end of the examination, indicate the number of sheets used in the appropriate box on
your cover sheet.
Unless otherwise stated in the question, all numerical answers must be given exactly or correct
to three significant figures.
… MOCK EXAM ……… MOCK EXAM ……… MOCK EXAM …
IB Math HL & SL workshop
Page 47
–2–
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: 5]
When the function f x 6 x4 11x3 22 x 2 ax 6 is divided by x 1 the remainder is 20 .
Find the value of a.
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IB Math HL & SL workshop
Page 48
–3–
2. [Maximum mark: 5]
A bag contains 2 red balls, 3 blue balls and 4 green balls. A ball is chosen at random from the
bag and is not replaced. A second ball is chosen. Find the probability of choosing one green ball
and one blue ball in any order.
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IB Math HL & SL workshop
Page 49
–4–
3. [Maximum mark: 6]
The 80 applicants for a Sports Science course were required to run 800 metres and their times
were recorded. The results were used to produce the following cumulative frequency graph.
Estimate
(a) the median;
[2 marks]
(b) the interquartile range.
[4 marks]
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IB Math HL & SL workshop
Page 50
–5–
4. [Maximum mark: 6]
4
2
Find the coordinates of the point where the line with the vector equation r 2 1
2
3
intersects the plane with the equation 2 x 3 y z 2 .
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IB Math HL & SL workshop
Page 51
–6–
5. [Maximum mark: 7]
(a) Express the complex number 8i in polar form.
[3 marks]
(b) The cube root of 8i which lies in the first quadrant is denoted by z. Express z
(i) in polar form;
[2 marks]
(ii) in Cartesian form.
[2 marks]
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IB Math HL & SL workshop
Page 52
–7–
6. [Maximum Mark: 7]
Find the equation of the line that is tangent to the curve 3x 2 4 y 2 7 where x 1 and y 0 .
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IB Math HL & SL workshop
Page 53
–8–
7. [Maximum mark: 6]
Find the value of x satisfying the equation
3 4 6
x
2 x 1
Give your answer in the form
x2
ln a
where a, b .
ln b
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IB Math HL & SL workshop
Page 54
–9–
8. [Maximum Mark: 6]
The independent events A and B are such that P A 0.4 and P A B 0.88 . Find
(a)
P B ;
[4 marks]
(b)
the probability that either A occurs or B occurs, but not both.
[2 marks]
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IB Math HL & SL workshop
Page 55
– 10 –
9. [Maximum Mark: 6]
The area of the enclosed region shown in the diagram is defined by
y x2 2, y ax 2, where a 0
y
2
0
a
x
The region is rotated 360 about the x-axis to form a solid of revolution. Find, in terms of a, the
volume of this solid of revolution.
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IB Math HL & SL workshop
Page 56
– 11 –
10. [Maximum Mark: 6]
The diagram below shows the graph of equation y1 f x , 0 x 4 .
On the axes below, sketch the graph of y2 f t dt , marking clearly the points of inflexion.
x
0
IB Math HL & SL workshop
Page 57
– 12 –
Section B
Answer all the questions on the answer sheets provided. Please start each question on a new page.
11. [Maximum Mark: 17]
The probability density function of the random variable X is given by
k
, for 0 x 1
f x 4 x2
0,
otherwise
(a) Find the value of the constant k.
(b) Show that E X
6 2 3
(c) Find the median of X.
[5 marks]
[7 marks]
[5 marks]
12. [Maximum Mark: 16]
(a) Find the root of the equation e22 x 2e x giving the answer as a logarithm.
[4 marks]
(b) The curve y e22 x 2e x has a minimum point. Find the coordinates of
this minimum.
[7 marks]
(c) The curve y e22 x 2e x is shown below.
Write down the coordinates of the points A, B and C.
(d) Hence state the set of values of k for which the equation e22 x 2e x k
has two distinct roots.
IB Math HL & SL workshop
[3 marks]
[2 marks]
Page 58
– 13 –
13. [Maximum Mark: 13]
(a) Show that the following system of equations will have a unique solution
when a 1 .
x 3y z 0
3x 5 y z 0
[5 marks]
x 5 y 2 a z 9 a2
(b) Given that a 1 , state the solution in terms of a.
(c) Hence, solve
x 3y z 0
3x 5 y z 0
[6 marks]
[2 marks]
x 5y z 8
14. [Maximum Mark: 14]
(i)
Using mathematical induction, prove that
n
r 12
r 1
r 1
n 2n
[7 marks]
(ii) The first three terms of a geometric sequence are also the first, eleventh and
sixteenth term of an arithmetic sequence.
The terms of the geometric sequence are all different.
The sum to infinity of the geometric sequence is 18.
(a) Find the common ratio of the geometric sequence, clearly showing
all working.
[4 marks]
(b) Find the common difference of the arithmetic sequence.
[3 marks]
IB Math HL & SL workshop
Page 59
… MOCK EXAM ……… MOCK EXAM ……… MOCK EXAM …
MATHEMATICS HIGHER LEVEL PAPER 1
► MARKSCHEME ◄
1.
[5 marks]
2.
[5 marks]
(C5)
IB Math HL & SL workshop
Page 60
3.
4.
(M1)
(A1)
[6 marks]
(C6)
5.
(A1)
IB Math HL & SL workshop
Page 61
(C3)
6.
(A2)
(A2)
Finding equation of line; slope is
y 1
3
x 1
4
3
7
y x
4
4
3
and passes through 1,1
4
(M1)
(M1)
(A1)
[7 marks]
7.
[6 marks]
IB Math HL & SL workshop
Page 62
8.
[6 marks]
9.
(M1)
(A1)
[6 marks]
10.
[6 marks]
IB Math HL & SL workshop
Page 63
11.
[5 marks]
Total [17 marks]
IB Math HL & SL workshop
Page 64
12.
IB Math HL & SL workshop
Page 65
13.
IB Math HL & SL workshop
Page 66
14.
[4 marks]
continued on next page …
IB Math HL & SL workshop
Page 67
14. (continued)
[4 marks]
[4 marks]
Total [14 marks]
IB Math HL & SL workshop
Page 68
… MOCK EXAM ……… MOCK EXAM ……… MOCK EXAM …
MATHEMATICS
HIGHER LEVEL
PAPER 2
Name
Friday 18 March 2011
Candidate session number
0
2 hours
0
INSTRUCTIONS TO CANDIDATES
Write your name and session number in the boxes above.
Do not open this examination paper until instructed to do so.
A graphic display calculator is required for this paper.
Section A: answer all of Section A in the spaces provided.
Section B: answer all of Section B on the answer sheets provided. Write your session number
on each answer sheet, and attach them to this examination paper and your cover
sheet using the tag provided.
At the end of the examination, indicate the number of sheets used in the appropriate box on
your cover sheet.
Unless otherwise stated in the question, all numerical answers must be given exactly or correct
to three significant figures.
… MOCK EXAM ……… MOCK EXAM ……… MOCK EXAM …
IB Math HL & SL workshop
Page 69
–2–
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: 6]
Triangle ABC has C 42 , BC 1.74 cm, and area 1.19 cm 2 .
(a) Find AC.
[2 marks]
(b) Find AB.
[4 marks]
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IB Math HL & SL workshop
Page 70
–3–
2. [Maximum mark: 6]
Find the values of a and b, where a and b are real, given that a bi 2 i 5 i
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IB Math HL & SL workshop
Page 71
–4–
3. [Maximum mark: 6]
The function f is defined as f x
3x 4
, x 2.
x2
(c) Find an expression for f 1 x .
[5 marks]
(d) Write down the domain of f 1
[1 marks]
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IB Math HL & SL workshop
Page 72
–5–
4. [Maximum mark: 6]
The function f is defined as f x sin x ln x for x 0.5,3.5 .
(a) Write down the x-intercepts.
[2 marks]
(b) The area above the x-axis is A and the total area below the x-axis is B.
If A kB , find k.
[4 marks]
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IB Math HL & SL workshop
Page 73
–6–
5. [Maximum mark: 6]
The weights in grams of bread loaves sold at a supermarket are normally distributed with
mean 200 grams. The weights of 88% of the loaves are less than 220 grams.
Find the standard deviation.
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–7–
6. [Maximum mark: 6]
Find e2 x sin x dx .
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7. [Maximum mark: 6]
The number of car accidents occurring per day on a highway follows a Poisson distribution
with mean 1.5.
(a) Find the probability that more than two accidents will occur on a given day.
[3 marks]
(b) Given that at least one accident occurs on another day, find the probability that
more than two accidents occur on that day.
[3 marks]
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–9–
8. [Maximum mark: 6]
There are 10 seats in a row in a waiting room. There are six people in the room.
(a)
In how many different ways can they be seated?
[2 marks]
(b)
In the group of six people, there are three sisters who must sit next to each other.
In how many different ways can the group be seated?
[4 marks]
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– 10 –
9. [Maximum mark: 6]
Solve the differential equation
x 2
2
dy
4 xy
dx
x 2
given that y 1 when x 1 .
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10. [Maximum mark: 6]
The radius and height of a cylinder are both equal to x cm. The curved surface area of the
cylinder is increasing at a constant rate of 10 cm2 sec . When x 2 , find the rate of change of
(a) the radius of the cylinder,
[4 marks]
(b) the volume of the cylinder.
[2 marks]
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– 12 –
Section B
Answer all the questions on the answer sheets provided. Please start each question on a new page.
11. [Maximum mark: 12]
A machine is set to produce bags of salt, whose weights are distributed normally, with a
mean of 110 grams and standard deviation of 1.142 grams. If the weight of a bag of salt is
less than 108 grams, the bag is rejected. With these settings, 4% of the bags are rejected.
The settings of the machine are altered and it is found that 7% of the bags are rejected.
(a) (i) If the mean has not changed, find the new standard deviation, correct to three
decimal places.
[4 marks]
The machine is adjusted to operate with this new value of the standard deviation.
(ii) Find the value, correct to two decimal places, at which the mean should be set so
that only 4% of the bags are rejected.
[4 marks]
(b) With the new settings from part (a), it is found that 80% of the bags of salt have a weight
which lies between A grams and B grams, where A and B are symmetric about the mean.
Find the values of A and B, giving your answers correct to two decimal places.
[4 marks]
12. [Total mark: 22]
Part A
[Maximum mark: 12]
A bag contains a very large number of ribbons. One quarter of the ribbons are yellow
and the rest are blue. Ten ribbons are selected at random from the bag.
(a) Find the expected number of yellow ribbons selected.
[2 marks]
(b) Find the probability that exactly six of these ribbons are yellow.
[2 marks]
(c) Find the probability that at least two of these ribbons are yellow.
[3 marks]
(d) Find the most likely number of yellow ribbons selected.
[4 marks]
(e) What assumption have you made about the probability of selecting a yellow
ribbon?
[1 mark]
Part B
[Maximum mark: 10]
The continuous random variable X has probability density function
x
,
f x 1 x 2
0
for 0 x k
otherwise
(a) Find the exact value of k.
[5 marks]
(b) Find the mode of X.
[2 marks]
(c) Calculate P 1 X 2 .
[3 marks]
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– 13 –
13. [Total mark: 26]
Part A
[Maximum mark: 14]
(b) The line L1 passes through the point A 0,1, 2 and is perpendicular to the plane
x 4 y 3z 0 . Find a Cartesian equation of L1 .
[2 marks]
(b) The line L2 is parallel to L1 and passes through the point P 3, 8, 11 . Find the
vector equation of the line L2
[2 marks]
(c) (i) The point Q is on the line L1 such that PQ is perpendicular to L1 and L2 .
Find the coordinates of Q.
(ii) Hence find the distance between L1 and L2 .
Part B
[10 marks]
[Maximum mark: 12]
Consider the system of equations
x 2 y kz 0
x 3y z 3
kx 8 y 5 z 6
(a) Find the set of values of k for which this system of equations has a unique
solution.
[6 marks]
(b) For each value of k that results in a non-unique solution, find the solution set. [6 marks]
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33 Exam tips / advice / information
Maths SL & HL
1. Do not give up too easily on a question. Sometimes it is a good idea to move on to another
question and return later to one that you found difficult.
2. Time management / pacing is very important during the examination and too much time spent on a
difficult question may mean that you do not have enough time to complete other questions.
3. If you make a mistake draw a single line through the work that you want to replace. Do not cross
out the work until you have replaced it with something you consider better.
4. Include relevant explanations / notes with your algebraic work.
5. Do not try to save time by taking algebraic shortcuts. Be careful with algebraic manipulations.
6. It is best to make diagrams and graphs large – clearly drawn with all appropriate labels.
7. If an exact answer is not too difficult or tedious to obtain then it is best to give the exact answer.
Otherwise give an approximate answer to an accuracy of three significant figures.
8. Check for key words/phrases – such as “hence”, “show that”, “write down”, etc. Underlining key
words/phrases when you first read a question can be helpful.
9. Questions with the phrase “show that” give you the answer and ask you to show how the answer is
obtained from the given information – be clear and complete. Do NOT work backwards – that is,
do not start with the answer and work back to the given information. Since the answer is given to
you, your work will be marked only for ‘method’ and ‘reasoning’. Hence, you should expect to
include some words of explanation with your working.
10. Exam questions are often written in a way so that, even if you cannot get the answer to one part,
you can still answer later parts. This is especially true for Section B (extended-response
questions).
11. Be completely familiar with the course Information Booklet (formula booklet). You are given a
‘new’ copy of it when you take the exams. You cannot take your own copy into the exam.
12. When answering a trig question, be sure to check that your calculator is set to the correct angle
mode – degrees or radians.
13. Be familiar with common error messages that your calculator may display.
14. Graphs displayed on a GDC may be misleading – so make sure that what you see makes sense.
15. Be sure your GDC has new batteries for the exam.
16. Questions in each section (Section A and Section B) are roughly in order of difficulty. Therefore,
questions #1 and #2 in Section A should be easier than the questions near the end of Section A.
Likewise, the first question in Section B should be easier than the last question in Section B.
17. Questions in Section A (short-response questions) are generally testing you on only one or two
syllabus items.
18. Questions in Section B (extended-response questions) are generally testing you on two or more
syllabus items. The question will always have multiple parts and often results from one part may
be needed in a later part.
19. Since you are not allowed a GDC on Paper 1 the questions on this exam will focus on analytic /
algebraic / ‘thinking’ solutions. Be extra careful with arithmetic and algebraic computations
because you’re not able to do a check with your GDC.
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20. On Paper 2, if you solve an equation by means of a graph on your GDC you must provide a
clearly labelled sketch of the graph in your work – and indicate exactly what equation you solved
on your GDC.
21. Even though a GDC is “required” on Paper 2, do NOT assume that you will need to use your GDC
on every question on Paper 2; or every part of every question.
22. Do NOT use any calculator notation in your written solution. For example, if you are computing
the value of the derivative of function f for x 2 on your GDC, do NOT write
dy
nDeriv(Y£,X,3)=6. You should write f 3 6 or
6 when x 3 . Other examples
dx
include not writing down calculator commands for statistical computations, such as normalcdf
and invNormal.
23. If you do use your GDC to obtain an answer for a question on Paper 2, be sure that you clearly
write down the appropriate mathematical ‘set-up’ for the computation you will perform on your
GDC.
24. There will definitely be some questions on Paper 2 where it will be more efficient and easier to
find the answer by using your GDC rather than an analytic/algebraic method. After reading a
question on Paper 2, you need to ask yourself is whether using a GDC is appropriate in helping to
find the solution. Do not lose valuable time by choosing to answer a question using a tedious
analytic method when you could get the answer quickly by using your GDC.
25. Examples where a GDC would be needed include finding the inverse of a 3 × 3 matrix or
obtaining the standard deviation for a set of data. Examples of questions that will NOT appear on
Paper 1 include calculations of binomial coefficients in algebra, and statistics questions requiring
the use of tables.
26. In trigonometry, you are expected to be familiar with the characteristics of the sin, cos and tan
curves (include reciprocal trig functions for HL), including knowledge of the sin, cos & tan ratios
of 0 , 30 , 45 , 90 and multiples of these values.
6
4
2
27. Not all questions on Paper 2 will necessarily require the use of the GDC. There will be questions
where a GDC is not needed and others where its use is optional. There will be some questions that
cannot be answered without a GDC.
28. Do not skip doing easy simplifications. For example, do not leave an answer of 62 or sin 90 .
29. It is very important to clearly show your work for each question on both Paper 1 and Paper 2.
Correct answers without sufficient working may not earn full marks. Trying to use a correct
method, even if you do not get the correct answer, will often earn you some method marks.
30. Your exam paper is marked by a teacher (hired by the IB as an ‘examiner’). The examiner is
looking to give you marks, so make his/her job easier by writing your solutions clearly and
completely. It’s always worth making an attempt to show some reasonable work for a question.
You cannot lose marks for incorrect work.
31. Your work for questions in Section A must be written on the same page as the question. You may
continue your solution below the dashed lines.
32. Your work for questions in Section B must be written on separate answer sheets. You must start
the solution for each question in Section B on a new answer sheet.
33. Eat breakfast !
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~ Diversion #2 ~
Two algebraic puzzles
► Proof that 2 1 ◄
Let
ab
Multiply both sides by a:
a 2 ab
Subtract b 2 from both sides by a:
a 2 b2 ab b2
Factorise:
a b a b b a b
Divide both sides by a b :
ab b
But a b ; therefore
2b b
Divide both sides by b:
2 1
► Proof that 1 0 ◄
x 1
2
x2 2 x 1
x 1 2 x 1 x2
2
x 1 2 x 1 x 2 x 1 x2 x 2 x 1
2
x 1 x 1 2 x 1 14 2 x 1
2
2
x 2 x 2 x 1 14 2 x 1
x 1 12 2 x 1 x 12 2 x 1
2
2
2
x 1 12 2 x 1 x 12 2 x 1
x 1 x
1 0
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Internal Assessment (first exams in 2014)
► The Mathematical Exploration ◄
Suggestions and guidance for implementation
(from draft of teacher support material – TSM)
The internally assessed component in these courses is a mathematical exploration. This is a
short report (6-12 pages) written by the student based on a topic chosen by him or her, and it
should focus on the mathematics of that particular area. The emphasis is on mathematical
communication (including formulae, diagrams, graphs and so on), with accompanying
commentary, good mathematical writing and thoughtful reflection. A student should develop
his or her own focus, with the teacher providing feedback via, for example, discussion and
interview. This will allow all students to develop an area of interest for them, without a time
constraint as in an examination, and will allow all to experience a feeling of success.
In addition to testing the objectives of the course, the exploration is intended to provide
students with opportunities to increase their understanding of mathematical concepts and
processes, and to develop a wider appreciation of mathematics. These are noted in the
aims of the course, in particular aims 6-9 (applications, technology, moral, social and ethical
implications, and the international dimension). It is intended that, by doing the exploration,
students benefit from the mathematical activities undertaken and find them both stimulating
and rewarding. It will enable students to acquire the attributes of the IB learner profile.
Contents
1.
Teacher responsibilities
2.
Skills and strategies required by students
3.
Developing the exploration
4.
Planning
5.
Frequently asked questions
6.
Internal Assessment criteria
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The Mathematical Exploration
1.
Teacher responsibilities
The teacher has 10 main responsibilities
During the process:
To advise students in choosing an appropriate topic for an exploration
To provide opportunities for students to learn skills related to exploration work
To ensure that students understand the assessment criteria and how they will be
applied
To encourage and support students throughout the research and writing of
explorations
To provide students with feedback
At the end of the process:
To verify the accuracy of all calculations
To assess the work accurately, annotating it appropriately to indicate where
achievement levels have been awarded
To ensure that the relevant form from the Handbook of procedures for the Diploma
Programme has been completed, justifying, with comments, the marks awarded
To ensure that the relevant form from the Handbook of procedures for the Diploma
Programme has been signed by both the student and the teacher, declaring that the
exploration is the student’s own work
To ensure that students fully understand the strengths and weaknesses of the
exploration
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The Mathematical Exploration
2.
Skills and strategies required by students
The exploration is a significant part of the course. It is useful to think of it as a developing
piece of work, which requires particular skills and strategies. As a general rule, it is
unrealistic to expect all students to have these specific skills and to follow particular
strategies before commencing the course.
Many of the skills and strategies identified below can be integrated into the course of study
by applying them to a variety of different situations both inside and outside the classroom. In
this way, students can practice certain skills and learn to follow appropriate strategies in a
more structured environment before moving on to working independently on their
explorations.
Choosing a topic
Identifying an appropriate topic
Developing a topic
Devising a focus that is well defined and appropriate
Ensuring that the topic lends itself to a concise exploration
Communication
Expressing ideas clearly
Identifying a clear aim for the exploration
Focusing on the aim and avoiding irrelevance
Structuring ideas in a logical manner
Including graphs, tables and diagrams at appropriate places
Editing the exploration so that it is easy to follow
Citing references where appropriate
Mathematical presentation
Using appropriate mathematical language and representation
Defining key terms, where required
Selecting appropriate mathematical tools (including information & communication
technology)
Expressing results to an appropriate degree of accuracy
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The Mathematical Exploration
Personal engagement
Working independently
Asking questions, making conjectures and investigating mathematical ideas
Reading mathematics and researching areas of interest
Looking for and creating mathematical models fo real-world situations
Considering historical and global perspectives
Exploring unfamiliar mathematics
Reflection
Discussing the implications of results
Considering the significance of the exploration
Looking at possible limitations and/or extensions
Making links to different fields and/or areas of mathematics
Use of mathematics
Demonstrating knowledge and understanding
Applying mathematics in different contexts
Applying problem-solving techniques
Recognizing and explaining patterns, where appropriate
Use of technology
One of the objectives for all group 5 subjects is to “use technology accurately, appropriately
and efficiently both to explore new ideas and to solve problems.”
The exploration may offer opportunities for this objective to be achieved, although this is not
a requirement for the exploration. For external assessment, the use of technology is limited
to the graphic display calculator, but for the exploration there are no such limitations. It is
reasonable, but not essential, to expect that the students, when producing their explorations,
will utilize technology in one or more ways.
Examples include:
Any kind of calculators, the internet, data logging devices
Word processing packages, spreadsheets, graphics packages
Statistics packages or computer algebra packages
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The Mathematical Exploration
3.
Developing the exploration
Although the exploration is likely to be written in the second year of the course, students
should be familiar with the concept of the exploration at a very early stage. The specific
planning and timing of the exploration will vary from school to school.
The following are suggestions that could be adopted at the different stages of the
exploration.
Before students start the exploration
Give out the criteria and stimuli early in the course and familiarize students with aims 6-9
Give notice of a time frame for doing the exploration
Encourage students to keep a record of ideas during the course (journal, notebook)
Encourage students to look for ideas everywhere (e.g. reading material), and give
access to such material (e.g. TV shows, websites, other courses)
Point out opportunities for exploring mathematics in everyday syllabus work
Give students opportunities to practise mathematical writing
Familiarize students with available technology
At the beginning of the exploration
Look at examples from the TSM or other students’ work
Brainstorm and/or use mind-mapping activities
Encourage the sharing of ideas
Ensure that students have a clear written focus before starting to write the exploration
While students are doing the exploration
Encourage self-assessment
Provide opportunities for discussion and questions
Provide appropriate feedback on the draft
After students have submitted the exploration
Ensure that internal standardization between teachers takes place, including between
mathematics SL and mathematics HL teachers
Discuss with students the strengths and weaknesses of their exploration
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The Mathematical Exploration
4.
Planning
Ensure that students have time to explore mathematics
Give a realistic deadline for submission of a draft of the written exploration
Give a realistic deadline for feedback to the students
Give a realistic deadline for final submission
Be aware of students’ mathematical experience in relation to the exploration at the
time of doing the exploration and record this
Long-term planning
The aim of long-term planning is to put the exploration into perspective in relation to the
whole course. It should take into account the following:
The sequencing of teaching units over the duration of the course
Those topics that are more applicable to the exploration
Appropriate places where the skills and strategies of the exploration can be
introduced
Opportunities for students to record and develop ideas relevant to the exploration; e.g.
journals or blogs
The resources available to them – both in and out of school
The role, if any, that the exploration will play in terms of school assessment
Timetabling exploration deadlines in the school calendar
Short-term planning
The aim of short-term planning is to provide a framework for the exploration so that students
gain the maximum benefit from the experience.
It is expected that teachers will give help and guidance to the students while they are doing
the exploration. Ten hours of class time should be allocated to management of the
exploration work. Some of this time can be taken up with individual or group activities,
where students learn some of the skills associated with exploration work. It is expected that
students will spend additional time working on their explorations outside class time.
Teachers should briefly discuss the exploration early during the course, so that students are
aware of what is required and that this is an essential part of the course.
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The Mathematical Exploration
A possible time frame for the exploration
It is envisaged that 10 hrs of class time and approximately 10 hrs outside class be spent on the exploration.
■ Choosing a focus/topic: 2 weeks
Class time: 2 – 3 hours
This will involve introductory lesson(s) leading to each student having a focused aim to their exploration. The
purpose and scope of the exploration should be explained. In doing this, teachers could demonstrate in various
ways how a stimulus will be used. The list below shows the wide range of stimuli that are suitable as starting
points to generate an idea as a focus for the exploration.
It could also be useful to look at an example of one or more stimuli and discuss with students how this could
lead to a focus for a mathematical exploration.
Examples of explorations from the TSM and other sources could be looked at to demonstrate to students what
is expected for them.
At the end of this period, each student should have decided on a focus for the exploration and have a
preliminary plan of how to approach it. This could involve describing the aims in class and inviting discussion
and comment from fellow students.
■ Draft exploration: 3 weeks
Class time: 4 – 5 hours
Class time could be used for writing the exploration (though it is envisaged that students will also spend time
outside lessons researching and writing their exploration and preparing a draft to submit). Discussion among
their peers and with the teacher is encouraged, but it is essential that the written draft exploration submitted is
the student’s own work and he or she should be prepared to explain any aspects of their work. Teachers may
also utilize this time to review and comment on drafts.
■ Teacher to review and comment on drafts: 4 – 8 weeks
Class time: 1 – 2 hours
This draft should be reviewed by the teacher and comments made on the strengths and weaknesses of the
work. This first draft must not be heavily annotated or edited by the teacher, but it is an opportunity for students
to receive further guidance on the exploration. This time could be used by students to consider and possibly
discuss the implications of this draft.
■ Final writing: 2 weeks
Class time: 1 – 2 hours
The student will have now have a short period in which to finalize the exploration based on the draft and the
advice given. During this time, the student can discuss their work with the teacher, but the final document must
be exclusively their own work. It is after this stage that the work will be marked by the teacher.
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The Mathematical Exploration
Stimuli
Students sometimes find it difficult to know where to start with a task as open-ended as this.
While it is hoped that students will appreciate the richness of opportunities for mathematical
exploration, it may sometimes be useful to provide a stimulus as a means of helping them to
get started on their explorations.
Possible stimuli the could be given to students include:
sport
archaeology
computers
algorithms
cell phones
music
sine
musical harmony
motion
e
electricity
water
space
orbits
food
volcanoes
diet
Euler
games
symmetry
architecture
codes
the internet
communication
tiling
population
agriculture
viruses
health
dance
play
pi ( )
geography
biology
business
economics
physics
chemistry
ITGS
psychology
Authenticity
Authenticity must be verified by signing the relevant form from the Handbook of procedures
for the Diploma Programme by both student and teacher.
By supervising students throughout, teachers should be monitoring the progress that
individual students are making and be in a position to discuss with them the source of any
new material that appears, or is referred to, in their explorations. Often, students are not
aware when it is permissible to use material written by others or when to seek help from
other sources. Consequently, open discussion in the early stages is a good way of avoiding
these potential problems.
However, if teachers are unsure as to whether an exploration is the student’s own work, they
should employ a range of methods to check this fact. These may include:
▪ Discussion with the student
▪ Asking the student to explain the methods used and summarize the results and conclusions
▪ Asking the student to replicate part of the analysis using different data
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The Mathematical Exploration
6.
Internal Assessment Criteria
The exploration is internally assessed by the teacher and externally moderated by the IB
using assessment criteria that relate to the objectives for mathematics SL. Each exploration
is assessed against the following five criteria. The final mark for each exploration is the sum
of the scores for each criterion. The maximum possible final mark is 20. Students will not
receive a grade for mathematics SL if they have not submitted an exploration.
A Communication (4 marks): This criterion assesses the organisation and coherence of the
exploration. A well-organised exploration has an introduction, a rationale (which includes a
brief explanation of why the topic was chosen), describes the aim of the exploration and has
a conclusion. A coherent exploration is logically developed and easy to follow.
B Mathematical Presentation (3 marks): This criterion assesses to what extent the
student is able to:
• Use appropriate mathematical language (notation, symbols & terminology)
• Define key terms, where necessary
• Use multiple forms of mathematical representation such as formulae, diagrams, tables,
charts, graphs and models
C Personal Engagement (4 marks): This criterion assesses the extent to which the
student engages with the exploration and makes it their own. These include thinking
independently and/or creatively, addressing personal interest and presenting mathematical
ideas in their own way.
D
Reflection (3 marks): This criterion assesses how the student reviews, analyses and
evaluates the exploration. Although reflection may be seen in the conclusion to the
exploration, it may also be found throughout the exploration. Reflection may be
demonstrated by consideration of limitations and/or extensions and relating mathematical
ideas to your own previous knowledge.
E
Use of Mathematics (6 marks): This criterion assesses to what extent students use
mathematics in the exploration. The mathematics explored should either be part of the
syllabus, or at a similar level or beyond. It should not be completely based on mathematics
listed in the prior learning. If the level of mathematics is not commensurate with the course,
a maximum of two marks can be awarded for this criterion. A piece of mathematics can be
regarded as correct even if there are a few minor errors so long as they do not cause a
disruption to the flow of mathematics or lead to an incorrect or inaccurate result.
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Internal Assessment Criteria (SL)
The Mathematical Exploration
A Communication
0
The exploration does not reach the standard described by the descriptors below.
1
The exploration has some coherence.
2
The exploration has some coherence and shows some organisation.
3
The exploration is coherent and well organised.
4
The exploration is coherent, well organised, concise and complete.
B
Mathematical Presentation
0
The exploration does not reach the standard described by the descriptors below.
1
There is some appropriate mathematical presentation.
2
The mathematical presentation is mostly appropriate.
3
The mathematical presentation is appropriate throughout.
C Personal Engagement
0
The exploration does not reach the standard described by the descriptors below.
1
There is evidence of limited or superficial personal engagement.
2
There is evidence of some personal engagement.
3
There is evidence of significant personal engagement.
4
There is abundant evidence of outstanding personal engagement.
D Reflection
0
The exploration does not reach the standard described by the descriptors below.
1
There is evidence of limited or superficial reflection.
2
There is evidence of meaningful reflection.
3
There is substantial evidence of critical reflection.
E
Use of Mathematics
0
The exploration does not reach the standard described by the descriptors below.
1
Some relevant mathematics is used.
2
Some relevant mathematics is used. Limited understanding is demonstrated.
3
4
5
6
Relevant mathematics commensurate with the level of the course is used. Limited
understanding is demonstrated.
Relevant mathematics commensurate with the level of the course is used. The
mathematics explored is partially correct. Some knowledge and understanding are
demonstrated.
Relevant mathematics commensurate with the level of the course is used. The
mathematics explored is mostly correct. Good knowledge and understanding are
demonstrated.
Relevant mathematics commensurate with the level of the course is used. The
mathematics explored is correct. Thorough knowledge and understanding are
demonstrated.
IB Math HL & SL workshop
Page 105
Mathematical Exploration HL/SL ~ Student Checklist
■ Is your report written entirely by yourself – and trying to avoid simply replicating
work and ideas from sources you found during your research?
Yes No
■ Have you strived to apply your personal interest; develop your own ideas; and use
critical thinking skills during your exploration and demonstrate these in your report? Yes No
■ Have you referred to the five assessment criteria while writing your report?
Yes No
■ Does your report focus on good mathematical communication – and read like
an article for a mathematical journal?
Yes No
■ Does your report have a clearly identified introduction and conclusion?
Yes No
■ Have you documented all of your source material in a detailed bibliography
in line with the IB academic honesty policy?
Yes No
■ Not including the bibliography, is your report 6 to 12 pages?
Yes No
■ Are graphs, tables and diagrams sufficiently described and labelled?
Yes No
■ To the best of your knowledge, have you used and demonstrated mathematics
that is at the same level, or above, of that studied in IB Mathematics HL/SL?
Yes No
■ Have you attempted to discuss mathematical ideas, and use mathematics, with a
sufficient level for IB Mathematics HL/SL?
Yes No
■ Are formulae, graphs, tables and diagrams in the main body of text?
(preferably no full-page graphs; and no separate appendices)
Yes No
■ Have you used technology – such as a GDC, spreadsheet, mathematics software,
drawing & word-processing software – to enhance mathematical communication? Yes No
■ Have you used appropriate mathematical language (notation, symbols,
terminology) and defined key terms?
Yes No
■ Is the mathematics in your report performed precisely and accurately?
Yes No
■ Has calculator/computer notation and terminology not been used?
Yes No
2
( y x , not y x ^ 2 ; , not for approx. values; , not pi; x , not abs(x); etc)
■ At suitable places in your report – especially in the conclusion – have you included
reflective and explanatory comments about the mathematical topic being explored? Yes No
IB Math HL & SL workshop
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Mathematics SL
Internal Assessment - *last exams 2013*
► Important information ◄
General
Every student must produce a portfolio containing two pieces of work (sometimes
referred to as a student report) completed during the course. Each piece of work (or report)
in the portfolio is internally assessed by the teacher against six criteria that are related to the
objectives of the mathematics SL course A sample of student portfolios from each school is
then externally moderated to ensure uniformity of standards. The portfolio is worth 20% of
the total score for the mathematics SL course.
Each portfolio task is assigned by the teacher. The tasks must be based on different areas of
the course and represent two types of tasks: mathematical investigation (type I) and
mathematical modelling (type II).
It is recommended that teachers set more than two tasks. For most teachers the goal should
be to assign two Type I tasks and two Type II tasks.
Generally speaking, teachers can choose to assign a task that is derived from three different
sources: (1) a task issued by the IB (only use tasks that have been identified to be used for
the appropriate exam session); (2) a task written by a teacher (most likely the teacher of the
students involved); or (3) an IB-issued task that has been modified by the teacher but still is
at an appropriate level of mathematics and allows students to successfully address all six
criteria.
Tasks taken from sources other than the set of IB tasks must be carefully reviewed to ensure
that they adequately meet the requirements of the tasks as described in the SL subject
guide, and that they offer students the opportunity to achieve at the highest level of each
criterion. To not do so make cause serious loss of marks on the IA scores of all students in
a school. It is critical that teachers completely work through any task before assigning
it to their students. This will help a teacher: (1) develop a set of notes (or marking key) to
guide them when scoring their students’ work; (2) anticipate difficulties that their students
may encounter; and (3) consider whether the instructions need to be clarified and/or
modified for their students.
The purpose of the portfolio
rewarded for mathematics carried out without the time limitations and pressure associated with written
examinations
increase their understanding of mathematical concepts and processes
hoped that students find the portfolio tasks both stimulating and rewarding
develop students’ personal insights into the nature of mathematics and to develop their ability to ask their
own questions about mathematics
allow them the students to experience the satisfaction of applying mathematical processes on their own
discover, use and appreciate the power of a calculator or computer as a tool for doing mathematics
develop the qualities of patience and persistence, and to reflect on the significance of the results they
obtain
for students to show, with confidence, what they know and what they can do
IB Math HL & SL workshop
Page 107
The level of sophistication of the students’ mathematical work should be similar to that
contained in the syllabus. It is not intended that additional topics are taught to students to
enable them to complete a particular task.
Portfolio tasks should be completed at intervals throughout the course and should not be left
until towards the end.
Portfolio work should be constructively integrated into the course of study so that it enhances
student learning by (1) introducing a topic, (2) reinforcing a topic that is currently being
taught meaning, or (3) reviewing a topic that was previously taught.
The need for proper mathematical notation and terminology, as opposed to calculator or
computer notation must be stressed and reinforced, as well as adequate documentation of
technology usage. Students will therefore be required to reflect on the mathematical
processes and algorithms the technology is performing, and communicate them clearly and
succinctly.
Type I – Mathematical Investigation
Essential skills:
Produce a strategy
Generate data
Recognize patterns or structures
Search for further cases
Form a general statement
Test a general statement
Justify a general statement
Appropriate use of technology
Type II – Mathematical Modelling
Essential skills:
Identify problem variables
Construct relationships between these variables
Manipulate data relevant to the problem
Estimating values of model parameters not measured or calculated from the data
Evaluating the usefulness of the model
Communicating the entire process
Appropriate use of technology
Level of tasks
Teachers should set tasks that are appropriate to the level of the course. In particular, tasks
appropriate to a standard level course, rather than to a higher level course, should be set.
IB Math HL & SL workshop
Page 108
Portfolio tasks – students working with teachers
Using previous student work
Consider giving students the opportunity to mark student work from previous years.
Writing good mathematics
Students should understand that the pieces of work they produce are for other people to
read. The mathematics they write should therefore be clear and logical, and contain
appropriate links and explanations. Students should endeavour to write a report that can be
read without a need to refer to the task instructions. Any mathematical results, data,
diagrams, etc should be embedded in the appropriate location in the report and not
separated from the relevant text (e.g. in an appendix at the end of the report)
Introductory tasks
If sufficient class time is available, it is highly recommended that a teacher assign a practice
portfolio task to introduce students to effective techniques and approaches for successfully
writing a mathematical report. The introductory task could include a piece of guided work
where class time is spent discussing each step. A practice task should be based on familiar
work, so that emphasis is on the process of completing a task. Whichever topic is chosen
for a practice task, it should be integrated into the course of study by being directly relevant
to the work that the students are currently being taught, rather than remaining an isolated
piece of mathematics.
It is also highly recommended that some class time is devoted to showing students the
different technological options that they have to choose from – both software and hardware
(including calculators) – for use in developing their mathematical ideas when addressing the
questions in a portfolio task.
Giving advice / feedback to students
Students are not expected to work in complete isolation on a task. In particular, teachers
should not try to reproduce examination conditions.
Some students may need extra encouragement to overcome initial difficulties and
misunderstandings, and teachers need not feel inhibited in giving advice to students. If
students ask specific questions, teachers should, where appropriate, guide them into
productive routes of inquiry rather than provide a direct answer.
If students do the mathematics themselves and write up their own findings
unaided then the work can be considered to be the students’ own.
Teachers need to provide feedback to students on the individual achievement levels
awarded for each criterion so that they can take action to improve their future performance.
It is therefore important that students are provided with copies of the assessment criteria and
are informed of the way in which achievement levels are awarded. Feedback to students
may also be provided through discussion with individuals, small groups or through wholeclass discussion. Teachers are encouraged to write feedback on the student work
submitted.
Ensuring that the work submitted is the student’s own
Students need to be aware from the beginning that any portfolio work submitted for
assessment must be entirely their own work.
IB Math HL & SL workshop
Page 109
Management of the Portfolio
Planning
As mentioned previously, portfolio work should be integrated into the course of study so that
it enhances student learning by introducing a topic, reinforcing a topic currently being taught
or reviewing a topic previously taught.
Integration of portfolio work will occur more naturally if teachers write their own tasks. If a
task written by someone else (including the IBO) is used, then adjustments may be needed
to ensure that the mathematics is relevant to the topic being taught and to the background of
the students.
Record keeping
It will be useful to record the following details at the time each task is set and assessed:
The areas of the syllabus on which the task is based
The date the task was given and the date of submission
The type of task (type I or type II)
Marking key (e.g. numerical / algebraic results; things to look for w.r.t. each criterion)
Background to the task in relation to what skills & concepts have been taught to students
Availability of technology
This information is needed so that moderators can be aware of the context in which a task
was set.
important
- Information / documents to go with moderation sample -
When marking student work, teachers are encouraged to write brief comments directly on
the student work to help explain why different achievement levels were awarded.
The moderator will need some kind of indication of how each achievement level was
awarded. It is recommended that this justification of achievement levels is recorded on
the reverse side of the form 5/PFCS, or in the optional ‘Feedback to Student’ form B.
The following documentation needs to be included with the moderation sample that is
given to your IBDP coordinator by your school’s internal deadline:
One completed copy of form 5/IA for the entire sample
One completed portfolio coversheet form 5/PFCS for each student in the sample –
signed and dated by the teacher and student
The portfolio for each student in the sample, containing two pieces of work
(originals, not photocopies)
A copy of each portfolio task (instructions) for which any student in the sample
submitted a report. Even if the task was issued by the IB you should still include a
copy of the task
Your set of Teacher’s Notes/Marking Key for each task
The background information listed in the Record Keeping section above (syllabus
area, date assigned and returned, task background, technology available). This
information can be entered in the optional ‘Teacher’s Record’ form A
IB Math HL & SL workshop
Page 110
Selecting the moderation sample
Teachers should ensure that they understand the requirements for submitting a sample.
Incomplete work should not be included – a portfolio containing fewer than two pieces of
work should not be part of the sample. If the sample selected by IBIS includes an
incomplete portfolio, another portfolio with the same (or similar) mark should be submitted as
well as the incomplete one.
Where there are two or more teachers of a subject within a school, they should agree on
standards before arriving at the final mark for each student. That is, internal
standardization of marks should take place within the school.
Time allocation
Approximately ten hours of classroom time should be allocated for four tasks (preferably)
that your students complete. This allows time to explain IA requirements, to make
clarifications to individual task instructions, answer group questions (when appropriate) and
some class time for students to work individually (with teacher supervision).
Setting of tasks
There is no requirement to provide identical tasks for all students (although recommended
since it’s easier for teacher to manage), nor to provide each student with a different task.
Teachers can design their own tasks, use those contained in published teacher support
material and the online curriculum centre (OCC), or modify tasks from other sources.
Submission of work
The finished piece of work should be submitted to the teacher for assessment about one to
two weeks after it has been set. Do not assign a portfolio task over an extended school
holiday. Students should not be given the opportunity to resubmit a piece of work after it
has been assessed.
Although highly recommended it is not required for work to be word-processed (equation
editor). However, if the work is not word-processed, it must be presented in ink.
Please note that when sending sample work for moderation, original work with teachers’
marks and comments on it must be sent. Photocopies are not acceptable.
Authenticity
Students need to be aware that the written work they submit must be entirely their own.
Teachers should try to encourage students to take responsibility for their learning, so that
they accept ownership of the work and take pride in it. When completing a piece of work
outside the classroom, students must work independently. Although group work can be
educationally desirable in some situations, it is not appropriate for the portfolio.
It is also appropriate for teachers to ask students to sign each task before submitting it to
indicate that it is their own work.
Incomplete portfolios
If only one piece of work is submitted, award zero for each criterion for the missing work.
Non-compliant portfolios
If two pieces of work are submitted and they are both type I or both type II, mark both tasks.
Apply a penalty of 10 marks to the final mark.
IB Math HL & SL workshop
Page 111
Portfolio Task ~ Student Checklist
■ Does your portfolio report read like an article for a mathematical journal?
Yes No
■ Can your portfolio report be read without referring to the task questions?
Yes No
■ Does the introduction give a brief overview of the report and its purpose?
Yes No
■ Does the conclusion summarize the findings?
Yes No
■ Has unnecessary repetition been avoided?
Yes No
■ Are graphs, tables and diagrams sufficiently labelled?
(including handwritten labels for calculator screen images)
Yes No
■ Do all graphs, tables and diagrams include a title and/or caption?
Yes No
■ Are graphs, tables and diagrams in the main body of text?
(preferably no full-page graphs; and no appendix containing graphs)
Yes No
■ Has technology been used in a way that helps develop mathematical ideas?
Yes No
■ Has the application of technology been clearly explained and/or demonstrated?
Yes No
■ Has calculator/computer notation and terminology not been used?
Yes No
2
( y x , not y x ^ 2 ; , not for approx. values; , not pi; x , not abs(x); etc)
■ Has the form declaring it is your own work been signed and included with report? Yes No
for Type I Tasks – Mathematical Investigation:
■ Has the general statement been clearly stated?
Yes No
■ Has the validity of the general statement been tested with further examples?
Yes No
■ Have comments on scope and limitations of the general statement been included? Yes No
(in main body of report and summarized in conclusion)
■ Is there a sufficient informal justification(SL)/proof(HL) of the general statement? Yes No
for Type II Tasks – Mathematical Modelling:
■ Are the variables, parameters and constraints clearly defined?
Yes No
■ Is there sufficient analysis of how well the mathematical model fits the data?
Yes No
■ Has the model been applied to other situations?
(“other situations” can include a change of parameter or more data)
Yes No
■ Is there discussion on the reasonableness of the results in the context of the task? Yes No
■ Are possible limitations & modifications of the results from the model discussed? Yes No
Task: ________________________________________
IB Math HL & SL workshop
Student: ____________________________
Page 112
IB Math HL & SL workshop
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IB Math HL & SL workshop
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Mathematics SL: The Portfolio
Form A
Teacher’s Record
Form A
Title of task:
Type:
I
II
Date Set:
Date submitted:
Syllabus topics covered
Background information
Purpose of the task
Previous exposure to relevant concepts/skills
Previous exposure to relevant terminology
Available technology
Teacher expectations regarding technology
IB Math HL & SL workshop
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Mathematics SL: The Portfolio
Form B
Feedback to student
Form B
Name:
I
II
Title of task:
Type:
Date Set:
Date submitted:
A.
Use of notation and terminology
/2
B.
Communication
/3
C.
Mathematical process
/5
D.
Results
/5
E.
Use of technology
/3
F.
Quality of work
/2
Total score:
IB Math HL & SL workshop
Page 116
/ 20
~ Diversion #3 ~
Two questions about chessboards
1.
How many squares are there on a chessboard?
2.
How many rectangles are there on a chessboard?
IB Math HL & SL workshop
Page 117
Algebra Preparation for Math SL/HL
25 Exercises (worked solutions on following pgs)
a b
2
x 4 3x 4 x 2 1
13. Simplify:
x2 4
2
k
1.
Expand:
2.
Simplify:
5 x4
3
6
15. solve for y:
x
x4
y2
3.
Simplify:
1
k
k 6 k 8
16. Solve for b:
a
6
bc b
4.
Solve for x:
5.
Expand:
m
2
14. solve for k: 7 3m
1
x
x 6 x 8
17. Solve for x:
x5 7
2
18. Solve for x:
x5 7 x
3
4 x h 4x
h
2
6.
7.
8.
9.
Simplify:
Simplify:
Simplify:
19. Solve for w: w4 3w2 4 0
2
20. Solve for x: x 5 9
a a
b c
c
21. Simply:
x
x
y
22. Solve for x:
x
x
y
x2
x2 5 x 5
2
23. Solve form m: 2m2 7m 4
Factorise: 3 p3 5 p 2 12 p
10. Factorise: 25m2 40m 16
5 x 1 3 x 2 x 1
2
11. Simplify:
5
x2
x2
x 1
3
24. Solve for x:
25. Is there anything incorrect with the
following solution ?
t 2 t 6 4 t 3
t 2 t 6 4 t 3 t 3 t 2 4 t 3
t 3 t 2
1
1
12. Simplify:
a b a b
IB Math HL & SL workshop
x3
1
x2
t 3
4 t 3
t 3
t24 t 2
therefore, the solution is t 2
Page 118
Worked Solutions – Algebra Preparation Exercises for Math SL/HL
1.
a b
2.
5 x 4 2 5 x 4 10 x 4 x 14
3
6
2 3
6
6
6
3.
1
k
1 k 8
k k 6
k 6 k 8 k 6 k 8 k 8 k 6
2
a 2 2ab b2
k 8
k 2 6k
k 2 7k 8
k 2 2k 48 k 2 2k 48 k 2 2k 48
the denominator factorises (numerator does not), but writing answer as
k 2 7k 8
is not
k 6 k 8
k 2 7k 8
considered simplifying (only factorise for a reason); best to leave simplified answer as 2
k 2k 48
4.
1
x
x x 6 x 8 x2 7 x 8 0
x 6 x 8
could solve this equation with quadratic formula, but since the quadratic expression x2 7 x 8
factorises then solving by factoring is preferred since it is more efficient (and less prone to errors)
x2 7 x 8 0
x 8 x 1 0
x 8 0 OR x 1 0 x 8 OR x 1
5.
m
2
2 m2 2 m4 4m2 4
3
m6 4m4 4m2 2m4 8m2 8 m6 6m4 12m2 8
6.
2
2
2
2
4 x h 4 x 2 4 x 2 xh h 4 x
4 x 2 8 xh 4h2 4 x 2 8 xh 4h2
h
h
h
h
h 8 x 4h
8 x 4h
h
IB Math HL & SL workshop
Page 119
a a
ac ab
ac ab
ac ab 1 ac ab
b c bc bc
bc
c
c
c
bc
c
bc 2
1
1
7.
8.
x xy
x xy
x 1 y
x xy
y2
y2
y y
y
2
xy 2 x 2 xy 2 x 2
y
xy 2 x 2
y
x y2 x
x
2
x
2
y
y
y2
y
x
x
y
y 1 y y y 2
y2 x
x y2
9. 3 p3 5 p 2 12 p p 3 p 2 5 p 12 p 3 p 4 p 3
10. 25m2 40m 16 5m 4 5m 4 5m 4
5 x 1 3 x 2 x 1
2
11.
x 1
12.
13.
3
5 x 5 3x 6
x 1
2
2
x 1 5 x 1 3 x 2
3
x 1
2 x 11
x 1
2
1
1
a b
ab
2b
2 2
a b a b a b a b a b a b a b
x 4 3x 4 x 2 1
x2 4
14. 7 3m
4 x 3x 2 4 x 2 4 x 2 4 x 4
x2 4
x2 4
x 2 x 2 x 2
x 2 x 2 x 2
2
2
k 7 3m 2 k
k
7 3m
IB Math HL & SL workshop
Page 120
15.
x
x 4 x y 2 x 4 x xy 4 y 2 x 8
y2
3x 8 xy 4 y
16.
3x 8
x4
a
6
6 b c ab 6b 6c ab 6b ab 6c
bc b
b 6 a 6c b
6c
6c
OR b
6a
a6
x 5 7 x 5 49 x 44
17.
18.
3x 8 y x 4 y
x5
2
7 x
2
x 5 49 14 x x 2 0 x 2 15x 44
x 4 OR x 11
but need to check answers (why?)
checking answers
x 4:
45 74
x 11 :
11 5 7 11
9 3
x 4 x 11 0
OK
16 4
NO
thus, the only solution is x 4
19. incorrect solution:
t 2 t 6 4 t 3 t 3 t 2 4 t 3
t 3 t 2
t 3
4 t 3
t 3
t24 t 2
therefore, the solution is t 2
correct solution:
t 2 t 6 4 t 3 t 2 t 6 4t 12 t 2 5t 6 0
t 2t 3 0
t 2 OR t 3
by dividing both sides of equation by t 3 the solution of t 3 was ‘lost’ in incorrect solution
20. let w2 t : w4 3w2 4 0 t 2 3t 4 0 t 4 t 3 0 t 4 OR t 3
substitute t back in for w2 :
w2 4 w 2
OR
w2 3
no real solution
therefore, there are two solutions w 2 OR w 2
IB Math HL & SL workshop
Page 121
21.
x 5 9 x 5 9 OR x 5 9
x 14 OR x 4
x2
22.
x2
23.
5
x2
x3
x 2
1
2
x2
5
x2
x2
x2
1
x3
x 2 x 232
x2 5 x 5
OR
x3
x2 x3 1
x2
x2 x2
1
x3
x 2
x2 5 5 x
3
x2 5
2
5 x
2
x2 5 25 10 x x2 10 x 20 x 2
24. 2m2 7m 4 2m2 7m 4 0
sign chart
-4
2m 1 m 4 0
1
2
m
2m – 1
m+4
(2m – 1)(m + 4)
25.
–
– 0
+
–
+
–
0
+
+
+
solution: m 4 or m
1
2
x 3 x 2
x3
x3
x3 x2
1
1 0
0
0
x2
x2
x2 x2
x2
5
0 x2 0 x 2
x2
IB Math HL & SL workshop
Page 122
► Set of 13 Unit Tests for Maths SL ◄
Unit 1 Test – Fundamentals
1.
Simplify each radical expression.
(a)
2.
10
8
(b)
12 27
(a) Find the equation of the line that passes through the points 2, 6 and 4, 2 . Write
the equation in the form y mx c , if possible.
(b) What is the exact distance between the two points given in (a)?
3 2 x2 y
3
3.
Simplify:
4.
Simplify:
5.
Completely factorize the expression: 3x 2 y 27 y
6.
Solve the system of equations using any method you wish.
2 x 3 y 5
3y y 3
4
3
4x y 6
8 x 3 y 13
9x
2x 1
x4
7.
Solve for x:
8.
Solve for k: 8 2 m 3k m 1 k
9.
Simplify:
10. Expand:
3m
4
m2 m4
2a 1
3
11. Solve for w in terms of n in the formula:
1 5w
3n w
n2
Bonus Questions
(1) Rationalize the denominator
6
. Give answer completely simplified.
2 7
(2) Find exact coordinates of the points where line y x
IB Math HL & SL workshop
1
and circle x 2 y 2 1 intersect.
2
Page 123
Unit 2 Test – Functions and Equations
Math SL
1. Consider the following equation for a parabola y x 2 8x 11
(a) Write the equation in ‘vertex form’, that is in the form y a x h k . .
2
(b) Write down the coordinates of the parabola’s vertex.
(c) Find the exact coordinates of the x-intercepts of the parabola.
2. Consider the function f x
2
x2
(a) State the domain and range for f x
(b) Find f
x .
1
(c) State the domain and range for f
3.
x
1
State the domain and range for the function.
f x
1
9 x2
4. Solve the following quadratic equation by the method of ‘completing the square’.
2 x2 12 x 5 0
3
1
, find the following – completely simplified.
8 and g x
x
x 1
1
(a) f g x
(b) g f x
(c) f g x
5.
Given f x
6.
The graph at right shows y g x
On the coordinate planes provided on the next two pages, sketch the following:
(a) g x 1
(b) 3g x
IB Math HL & SL workshop
(c) g 2 x
(d) g x
Page 124
6. (a)
(b)
IB Math HL & SL workshop
Page 125
6. (continued)
(c)
(d)
IB Math HL & SL workshop
Page 126
7.
Consider the function f x 2 x 3 4
(a) Sketch a graph of the function on the grid below.
y
10
5
- 10
-5
5
10
-5
- 10
(b) Find any x- or y-intercepts.
(c) State the domain and range.
(d) Does f have an inverse function, i.e. f
x ?
1
Why?
8. The following diagram shows the graph of y g x . It has minimum and maximum points at
1, 13
and 5, 1 , respectively.
(5, -1)
(-1, -13)
(a) Given that g is a cubic function, state the domain and range of g.
(b) What are the coordinates of the minimum and maximum points of y g x 5 ?
IB Math HL & SL workshop
Page 127
x
Unit 3 Test – Sequences and Series
Math SL
1. The first term of an arithmetic series is 15, the last term is 57 and the sum is 273 . Find the
number of terms in the series and the common difference.
2. Consider the infinite geometric series 486 162 54 18
(a) For this series, find the common ratio, giving your answer as a fraction in its simplest form.
(b) Find the eighth term of this series.
(c) Find the exact value of the sum of the infinite series.
3. (a) Write the following series using sigma notation: 8 13 18
63
(b) Find the exact sum for the series.
4. Expand 2x y
5
- and write in simplest form.
5. The first term of a geometric sequence is 48 and the third term is 27.
(a) Find the two possible values for the common ratio r.
(b) For each value of r, find the sum of an infinite number of terms of the sequence.
6. Write down the first three terms for each sequence that is given below in sigma notation.
10
(a)
3i 5
i 4
2
(b) 4
r 1 3
7
r 1
[do not need to find sum; just first three terms]
5
2
7. Find the coefficient of x 3 in the expansion of x3 .
x
8. A theatre has 20 rows of seats. There are 15 seats in the first row, 17 seats in the second row,
and each successive row of seats has two more seats in it than the previous row.
(a) Calculate the number of seats in the 20th row.
(b) Calculate the total number of seats.
9
9.
3
Find the term independent of x in the expansion 2x 2 .
x
Bonus Question
The consecutive terms of a geometric sequence are given by x 2, x 6 and 2 x 3 .
(a) Find the possible value(s) of x.
(b) For each value of x, state the value of the common ratio r and list the three terms.
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Unit 4 Test – Exponential and Logarithmic Functions
Math SL
Part I – No calculator (Questions 1 & 2 only)
1.
Find the value of the expression.
(a)
3
3
3
(b) 32 5
2. Solve for x in each equation.
(a) 27 x 9
x 12
(c) log8 4 x
(b) log x 16 2
Part II – Calculator allowed. Show any work in the space provided. Circle or underline answers.
3.
Write each expression as the logarithm of a single quantity.
(a) 4logb 2 logb 10
(b) 2log a 3log b log c
4. Approximate the value of the logarithm to three significant figures.
5. Solve for x:
log3 15
log b x
3
log b 10
6. State the domain and range of each function.
(a) f x ln x 1
(b) g x 2x 1
7. The population of Logburgh grows by 3.4% each year. How long will it take for the population
of Logburgh to double?. Give your answer to the nearest tenth of a year.
y
8.
Find the equation that is shown in the graph.
1
(9, 1)
(0, 0)
-2
2
4
6
8
10
-1
9.
Solve for x. Approximate answers to three significant figures.
(a) e2 x1 40
(b) 5log x 2 10
IB Math HL & SL workshop
(c) 4.17 2.36
x3
274.9
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x
10. A new radioactive substance is discovered called Balonium. It decays such that its half-life
is 500 years. The mathematical model for the amount A of Balonium that remains of an
t
1 500
initial amount A0 is given by A t A0
where t is the number of years.
2
(a) If the initial amount of Balonium is 10 grams, how much remains after 1600 years?
(b) How long (to the nearest whole year) will it take for 50 grams of Balonium to decay
down to just 1 gram?
(c) How long (to the nearest whole year) will it take for any amount of Balonium to decay so
that there is just 1% remaining of the initial amount?
11. (a) What is the inverse of the function y e x ?
(b) Sketch the graph of f x e x 1 on the grid below.
(c) The graph of y e x 1 has one asymptote. What is the equation of the asymptote?
(c) Sketch a graph of the inverse of f x e x 1 . Label it f 1 x . The graph of f
has one asymptote. What is the equation of the asymptote?
(d) Find the equation for f
1
x .
1
y
7
6
5
4
3
2
1
-5
-4
-3
-2
-1
-1
1
2
3
4
5
6
7
x
-2
-3
-4
-5
12. Solve for x. Give your answer exactly.
log3 x 1 log3 x 9 2
Bonus questions:
(1) Find b:
e x ln3 b x
(2) Solve for x:
log 2 x log x 2
IB Math HL & SL workshop
8
3
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x
Unit 6 Test – Trigonometric Functions and Equations
Math SL
1. Write down the domain, range, period and amplitude for each function.
(a) y 3cos 2 x
2
x
(b) y 2 sin
4
2. (a) Write the expression cos 2 x sin x in terms of sin x only.
(b) Solve the equation cos 2 x sin x 0 for the interval 0 x 2 ; give answer(s) exactly.
3. The graph of a sine function y a sin b x c d , where a, b, c and d are integers, is
graphed below. Write down the values of a, b, c and d.
y
1
-
4
4
2
3
4
5
4
3
2
7
4
2
9
4
x
-1
-2
-3
4. Find all solutions for cos 3x
5. Given that tan
(a) sin
1
in the interval 0 x . Give exact answer(s).
2
5
and that 0 , find the exact values of:
12
2
(b) cos
(c) sin 2
6. The average monthly precipitation for Aberdeen is modelled by the trigonometric function
P 15sin M 3 65 , P is precipitation in mm, and M is the month (Jan 1 , Feb 2,
6
(a) Which month has the least precipitation?
How many mm?
(b) Give two months that have the same average precipitation. How many mm?
7. (a) How many solutions does the equation sin 4 x
3
have in the interval 0 x ?
4
Provide a very brief justification for your answer.
(b) Give the solutions for sin 4 x
IB Math HL & SL workshop
3
, 0 x , approximately to 3 significant figures.
4
Page 131
)
8. Solve the equation 2sin 2 x cos x 1 0 , 0 x 2 . Give exact answer(s).
9. The depth of water, h meters, measured at a sea pier t hours after midnight is given by the
function
2
h a b cos
k
t , where a, b and k are constants.
The water is at a maximum depth of 21 meters at midnight and noon, and is at a minimum
depth of 13 m at 06:00 and at 18:00.
Write down the values of: (a) a
(b) b
(c) k
x
10. The diagram below shows the graph of f x 1 tan for 360 x 360 .
2
(a) Write down …
(i) the period of the function;
(ii) the value of f 90
(b) Solve f x 0 for 360 x 360
IB Math HL & SL workshop
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Unit 7 Test – Triangle Trigonometry
Math SL
1.
Find the possible length(s) of side AC of triangle ABC given that AB = 13.8 cm, BC = 10.7 cm
and angle A = 34.7 .
2.
Find the measure of the largest angle in a triangle whose sides are of length 7x, 10x and 12x.
3.
Find the two possible areas for triangle FGH given FG 13 cm, GH 11 cm and angle F 52
4.
is an obtuse angle and sin . Find the exact values of (a) cos , and (b) sin 2 .
5.
A ship sails from a harbour for 20 km on bearing of 25 and then continues due east for 18 km.
(a) How far will the ship have to sail to get back to the harbour by the shortest route?
(b) What will be the bearing of this return trip?
6.
The figure below right is a square pyramid. Each edge of the square base is 10 cm and each of
the lateral edges PR, PS, PT and PU are 12 cm. Find the measure of the angle between one of
the triangular faces and the square base.
2
3
P
12
S
12
R
cm
10
cm
cm
12 cm
U
1.5
10 cm
T
12 cm
7.
Find the area of the shaded region in the figure above.
8. Twins Anna and Tanya, who are both 1.75 m tall, both look at the top of Cleopatra’s Needle in
Central Park, New York City. Anna’s line of sight to the top makes an angle of 40 with the
horizontal and Tanya’s line of sight makes an angle of 50 with the horizontal. If they are
standing 7 m apart, how tall is the needle?
Bonus Questions
(1) Find the value of the cosine of the smallest angle, in terms of k, in the right triangle shown .
k +6
k +2
k +4
(2) Find the angle measure between two diagonals of a cube.
IB Math HL & SL workshop
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Unit 8 Test – Vectors
Math SL
(can be given as two separate tests)
Part I
1. The position vectors of the points A and B with respect to the origin O are 2i 3 j and i 5 j
respectively.
(a) Find the position vector of the point C such that AC 2 AB .
(b) Calculate the angle between the vectors AB and OB .
2. Find a vector of magnitude 8 units and is parallel to the vector 2i + 5j .
3. The angle between the vector a i 2 j and the vector b 2i m j is 45 .
Find the values of m.
4. Find the measure of the three interior angles of the triangle that has the following vertices:
A 6, 1 , B 2, 3 and C 2, 3 .
5. The diagram shows a triangle OAB with OA a and OB b . C is a point on AB such that
AC:CB 3:1. Express the following vectors in terms of a and b.
(a) AB
(b) AC
(c) CB
(d) OC
B
C
b
O
a
A
6. The points P, Q, R and S have position vectors 2i 3 j , 3i 8 j , 7i 6 j and 7i n j
respectively, where n is a constant. Find the value of n such that PR is perpendicular to QS .
4
22
2
7. Given that a , b and c
, find:
3
11
4
(a) a unit vector perpendicular to a;
(b) the value of the constants m and n such that m a n b c
(c) the measure of the angle between a vector in the direction of a and a vector in the direction
of b.
IB Math HL & SL workshop
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Unit 8 Test – Vectors (continued)
Part II
3
4
1. If a 2 and b 2 find c such that a 4c 2b .
4
1
2. Find the angle between the vectors p and q given that p 3i 4 j and q 5i 12 j . Give your
answer to the nearest tenth of a degree.
3. Find the value of m and the value of n so that the vectors 4i j 5k and mi n j 15k are
parallel.
4. A, B, C and D are four points with coordinates 2,3 , 5,7 , 11,10 and 8,6 respectively.
(a) Show that ABCD is a parallelogram.
(b) The point P divides BC in the ratio 1:2. Find the coordinates of P.
(c) AP intersects BD at point Q. Find the coordinates of Q.
(d) In what ratio does Q divide AP?
5. Find the vector equation of each line that passes through the pair of given points.
(a) two dimensions: 3, 7 and 1,3
(b) three dimensions:
2,5, 6
and 7, 1, 2
3
1
1
6. For a 2 , b 5 and c 0 , find the following:
4
3
5
(a) a 3b c
(b) b c
(d) a b
(e)
(c) a unit vector parallel to b
a b b
L
H
7. HJKL is a rectangle and M is the midpoint of JK .
Express each of the following vectors in terms of MH and ML .
J
(a) LH
(b) MJ
(c) JH
M
K
8. Points A, B and C have position vectors 3i 4 j , i 3 j and 7i 7 j respectively. Let D be a
point on the x-axis such that ABCD is a parallelogram.
(a) (i) Find BC .
(ii) Find the position vector of D.
(b) Find the angle between BD and AC .
IB Math HL & SL workshop
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Unit 9 Test – Differential Calculus (two parts – given as two separate tests)
Math SL
Part I: Fundamentals
1.
Find the derivative,
(a) y 7 3 x 2
2.
dy
, of each function.
dx
6
(b) y 5
x
(c) y 4 x
The line y 16 x 9 is a tangent to the curve y 2 x3 ax 2 bx 9 at the point 1, 7 . Find
the values of a and b.
3. The curve y 2cx3 4 x2 4 x 3 has a stationary point at x 1 . Determine the value of c.
4. Find the equation of the line tangent to the curve y x3 5x 2 where x 1 .
5. Given that the function f is defined as f x x3 3x 2 24 x 5 , (a) find the coordinates of
any stationary points and indicate whether they are a max, min or neither; (b) find the
coordinates of any inflexion points.
6. If y
2 x5 x3 7
dy
, find
.
2
x
dx
[hint: simplify
2 x5 x3 7
before differentiating]
x2
7. Find the gradient of the tangent to the curve y x3 5x 1 at the point where x 2 .
1
8. Find the equation of the tangent to the curve y 2 x 2 3x at the point , 2 .
2
9. Find the interval(s) in which the function f x 3x 2 12 x 5 is decreasing.
10. Consider the function h x x 4 5x3 6 x 2 4 x
1
and at x 2 .
4
1
(b) Determine whether h has a maximum, minimum or neither at x and at x 2 .
4
(c) The graph of h has two inflexion points. Find the coordinates of one of these inflexion
points such that x 1.
(a) Show that the graph of h has a horizontal tangent at x
11.
Consider the function g x x3 7 x 2 5x with a domain of x 0 .
(a) For which value of x does g have a maximum value (for the indicated domain)?
(b) What is the maximum value of g for x 0 ?
12.
An object is moving along the x-axis such that its displacement (x cm from the origin) at t
seconds is given by the function x t t 3 9t 2 27t , t 0 .
(a) When is the object stationary?
(b) What is the rate of change of the object (give appropriate units) at t 1 second?
(c) Determine when the object’s acceleration is negative, and when it is positive.
IB Math HL & SL workshop
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Unit 9 Test - Differential Calculus (continued)
Math SL
Part II: Further Techniques & Applications
1. Given y 1 x 2 , find
dy
dy
. Simplify the expression for
.
dx
dx
2. Consider the function f x e x sin x .
(a) Find the derivative of f with respect to x, i.e. find f x .
(b) On the interval 0 x 4 , find the exact value of x where f x 0 .
(c) For this value of x determine whether f has a maximum or minimum and find the exact
coordinates of this extreme point.
3. Find the equation of the line tangent to the curve y cos 2 x where x
4. Given the function y x2 ln x, x 0 , find
4
.
d2y
dy
and
and simplify the expressions for each.
dx 2
dx
5. Find the equation of the line tangent to y x ln x at the point where x e
6. Explain why the curve y
9
3x has no stationary points.
x
x
7. Find the line normal to the graph of y 2 cos where x . Express the line exactly.
2
8. Find the exact coordinates of the inflection point on the curve y x ln x 2 .
9. Find the maximum area of a rectangle inscribed in an isosceles right triangle whose hypotenuse
is 20 cm long.
h
10. A six-sided box with a total surface area of 500 cm 2 is
constructed such that the length of its rectangular base is
twice as long as its width.
x
2x
(a) If the width of the rectangular base is represented by x and the width by 2x , show that the
250 2 x 2
height, h, of the box can be expressed in terms of x as h
.
3x
(b) Hence, find an expression for the volume of the box in terms of x.
(c) Find the dimensions of the (length, width & height) for which the box will have a
maximum volume and calculate the maximum volume.
IB Math HL & SL workshop
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Unit 10 Test – Integral Calculus
1.
Find each of the following:
(a)
2.
3x 2 dx
(b)
3
6 x 1 dx
By using a suitable substitution, find each of the following:
(a)
3.
Math SL
x
1 x
2
(b)
dx
x sin x dx
2
An object moves in a straight line. At time t seconds the object’s velocity is given by
v t t sin t . When t 0 the displacement of the object is zero metres.
(a) Find an expression for the displacement s in terms of t.
(b) What is the total displacement of the object from t 0 to t 2 . Give exact answer.
(c) What is the total distance traveled by the object from t 0 to t 2 . Give exact answer.
4.
Find the area of the region enclosed by the curves y ln x, y e x and the vertical line x 4 .
5. (a) If y x 4 x , find
(b) Hence, evaluate
2
0
dy
and simplify.
dx
8 3x
dx .
4 x
6. Find the area of the region bounded by the graphs of y x 2 2 x 3 and y x 1 .
y
y = x ln x – x
7. Consider the function f x x ln x x, x 0 .
(a) The function has an x-intercept at the point A. Find
the exact x-coordinate of A.
(0,0)
A
(b) Find the area of the region bounded by the graph of f
and the x-axis.
8. Find the volume of the solid formed by revolving the
6
region bounded by y , x 2 and x 4 about the x-axis.
x
IB Math HL & SL workshop
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x
Unit 11 Test – Statistics
1.
Math SL
After an exercise class, a group of 24 people measured their heart rates. The data is below.
125
128
108
118
112
100
144
108
96
133
98
105
123
156
112
120
144
148
140
126
106
148
135
85
(a) Find the following:
(i) the lower quartile
(iv) the interquartile range
2.
(ii) the median
(iii) the upper quartile
(v) the range of the data
The cumulative frequency graph below shows the heights of 120 girls in a school.
(a) Using the graph
(i) write down the median;
(ii) find the interquartile range.
(b) Given that 60% of the girls are taller
than a cm, find the value of a.
3.
Consider the data set {k − 3, k +1, k + 2, k + 4}, where k
.
(a) Find the mean of this data set in terms of k.
(b) Find the variance of this data set in terms of k.
Each number in the above data set is now decreased by 2.
(c) Find the mean of this new data set in terms of k.
(d) Find the variance of this new data set in terms of k.
IB Math HL & SL workshop
(continued on next page)
Page 139
4.
5.
In a sample of 50 boxes of light bulbs, the number of defective light bulbs per box is
shown below.
Number of defective light bulbs per box
0
1
2
3
4
5
6
Number of boxes
7
3
15
11
6
5
3
(a)
Calculate the median number of defective light bulbs per box.
(b)
Calculate the mean number of defective light bulbs per box.
The following is the cumulative frequency diagram for the heights of 30 plants given in
centimetres.
cumulative frequency
30
25
20
15
10
5
5
10
15
20
25
height (h)
(a)
Use the diagram to estimate the median height.
(b) Complete the following frequency table.
Height (h)
Frequency
0h5
4
5 h 10
9
10 h 15
15 h 20
20 h 25
(c)
6.
Hence estimate the mean height.
A fair six-sided die, with sides numbered 1, 1, 2, 3, 4, 5 is thrown. Find the mean and
variance of the score.
(continued on next page)
IB Math HL & SL workshop
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7.
Consider the six numbers, 2, 3, 6, 9, a and b. The mean of the numbers is 6 and the
variance is 10. Find the value of a and of b, if a < b.
8.
A machine produces packets of sugar. The weights in grams of thirty packets chosen at
random are shown below.
Weight (g) 29.6 29.7 29.8 29.9 30.0 30.1 30.2 30.3
Frequency
2
3
4
5
7
5
3
1
Find unbiased estimates of
(a)
the mean of the population from which this sample is taken;
(b)
the variance of the population from which this sample is taken.
9. Eight IB students were asked how many hours per month they studied Mathematics SL. The
results, along with their scores on a final course exam, are given in the table below.
Hours
Exam score
20
95
16
90
21
85
10
83
15
77
19
62
13
58
11
49
(a) Construct a scatter plot of the data.
(b) Write down the regression equation that will enable you to predict a student’s exam score
from the number of hours they study per month. Draw the line on your scatter plot.
(c) Interpret the slope of this line. Find the product-moment correlation coefficient r and
comment on the association between exam score and number of hours studied per month.
(d) What exam score would you expect for a student that studies 14 hours per month?
IB Math HL & SL workshop
Page 141
Unit 12 Test – Probability
1
Math SL
The Venn diagram below shows the number of students in a particular class who play
basketball, B, and the number of students in the class that play volleyball, V.
V
B
11
8
4
2
If a student is chosen at random from the class, find the probability that the student …
(a) plays basketball;
(b) does not play volleyball;
(b) plays at least one of the two sports;
(d) plays volleyball if it is know that the student plays basketball.
A coin is biased (i.e. not fair) so that P head
2
2
1
and P tail . If the coin is tossed six
3
3
times find the probability of obtaining
(a) exactly five heads;
3
(b) at least two heads;
(c) at least one tail.
Four different numbers are chosen at random from the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9. Find the
probability that the numbers chosen will be
(a) all odd;
4.
(b) two odd and two even numbers.
Given A and B are independent events, and that p A 0.64, p A B 0.73, find p B .
5. A box contains 36 apples of which 7 are rotten. If 8 apples are chosen at random, what is the
probability that
(a) none of them are rotten;
6.
(b) exactly one of them is rotten?
A bag contains twelve marbles. Six of the marbles are blue, four are red and two are green. If
two marbles are chosen (without replacement), what is the probability that:
(a) two blue marbles;
(b) one red marble and one green marble, in any order;
(c) two different coloured marbles.
(continued on next page)
IB Math HL & SL workshop
Page 142
7.
8.
Louis and Pierre play a match consisting of five games, each of which must be won or lost. In
2
each of the first three games the probability that Louis will win is
and in the remaining two
3
3
games the probability is . Find the probability that Louis will win four or more of the games.
4
If P X 0.8, P Y 0.6 and X and Y are independent events, find the probability that:
(a) both X and Y occur;
(b) X or Y occur but not both X and Y;
(b) X occurs given that Y did not occur;
(d) neither X nor Y occurs.
9.
Carlo travels to work by train every weekday from Monday to Friday. The probability that he
catches the 08.00 train on Monday is 0.66. The probability that he catches the 08.00 train on
any other weekday is 0.75. A weekday is chosen at random.
(a) Find the probability that he catches the train on that day.
(b) Given that he catches the 08.00 train on that day, find the probability that the chosen day is
Monday
IB Math HL & SL workshop
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Unit 13 Test – Probability Distributions
Math SL
1. A zoologist knows that the lengths of a certain type of tropical snake are normally distributed
with mean length L meters and standard deviation of 0.12 meters. If 20% of the snakes are
longer than 0.7 meters, find the value of L.
2. Juan always practices a set of three penalty kicks at the end of each football training session.
The discrete random variable X represents the number of successful penalty kicks that Juan
makes each time. The table below shows the probability distribution.
x
P X x
0
1
2
3
0.05
k
0.5
0.3
(a) Find the value of k.
(b) Find the expected number of successful penalty kicks Juan makes each training session.
3.
The scores for a national exam is scaled such that the mean score is 500 and the standard
deviation is 125. Given that the scores are normally distributed, what proportion of students
scored between 650 and 700?
4.
In order for a student to pass a mathematics course they must earn a passing score on at least
eight of ten tests. For each test the probability Joe passes is 0.86. Find the probability that Joe:
(a) passes exactly eight of the ten tests;
(b) passes the course.
5.
The volumes of a large number of bottles of a certain brand of juice drink are measured. They
are normally distributed with a mean volume of 310 ml and a standard deviation of 5 ml.
(a) What percentage of bottles contain between 300 and 310 ml?
(b) What percentage of bottles contain at least 304 ml?
(c) What is the probability of a bottle containing less than 300 ml?
6. The discrete random variable X has the following probability distribution.
k
, x 1, 2, 3, 4
P X x x
0, otherwise
(a) Find the value of the constant k;
(b) Find E x
7. It is known that 20% of the potatoes in a large stock are rotten. A random sample of 12
potatoes is to be taken. Find the probability that this sample will contain:
(a) exactly three rotten potatoes;
(b) at least four rotten potatoes.
IB Math HL & SL workshop
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~ Diversion #4 ~
Four Spheres
Three spheres each with a radius of one unit are placed on a table such that each
sphere is touching the other two. A fourth sphere of the same size is stacked on top of
the three spheres in an efficient manner such that each sphere is touching the other
three. What is the (shortest) distance from the table to the top of the fourth sphere (top
sphere)? Give the distance exactly.
IB Math HL & SL workshop
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~ Diversion #5 ~
Two Coins – but not both fair
Russian two-ruble coin issued in 2007 to commemorate
the 300th anniversary of the birth of Leonhard Euler
You have two coins in your pocket. One of them is double-headed and the other is a
fair coin (head-tail). You randomly choose one of the coins from your pocket. You
look at one side of it. Given that you see 'heads', what is the probability that you've
chosen the fair coin?
IB Math HL & SL workshop
Page 146
Theory of Knowledge
Mathematics
All is number. (Pythagoras)
There is little doubt that mathematics has influenced our understanding of the natural world. It has
been said that the world is mathematical at its deepest level. Whether mathematical knowledge
comes to us as a result of some connection to natural phenomena is another matter. Many have
likened mathematics to a logical game invented by humans. Others consider mathematics to be a
unique aesthetic experience, while still others consider it a 'special language tool'. The following
questions offer an opportunity to reflect on the nature of mathematical knowledge, which Diploma
programme students encounter in their Group 5 subject(s).
Definition of Mathematics
What does calling mathematics a 'language' mean? Does mathematics function in the
same way as our daily written and spoken language?
Do mathematical symbols have meaning, in the same sense as words have meaning?
Why is it that some claim that mathematics is no more than a 'logical game', such as
chess, for example, devoid of particular meaning? If this were the case, how do we
account for the fact that it seems to apply so well to the world around us?
What could Carl Sandburg have meant by the following?
'Arithmetic is where the answer is right and everything is nice and you can look out of
the window and see the blue sky – or the answer is wrong and you have to start all
over and try again and see how it comes out this time.'
Mathematics and Reality
Is it reasonable to claim that mathematics is effective in accounting for the workings of the
physical world?
Could it be argued that mathematics is simply the application of logic to questions of
quantity and space?
What did Einstein mean by asking: 'How can it be that mathematics, being after all a
product of human thought which is independent of experience, is so admirably appropriate
to the objects of reality?'
What are the differences between the formal school of thought which regards
mathematics as similar to an activity governed by rules, limited only by the rules of logic
and the creativity of the mathematician, and the realist school of thought which regards
mathematics as referring to the way the world actually works?
What is the foundation on which mathematical knowledge rests? Is it discovered or
invented? What is meant by this distinction? Can it be applied usefully in other areas?
What is the origin of the axioms of mathematics? Are axioms necessarily self-evident to all
people? How is an axiomatic system of knowledge different from, or similar to, other
systems of knowledge?
Do different geometries (Euclidean and non-Euclidean) refer to or describe different
worlds?
Mathematics and Knowledge Claims
What is the significance of proof in mathematical thought? Is a mathematical statement
true only if it has been proved? Is the meaning of a mathematical statement dependent on
its proof? Are there such things as true but unprovable statements in mathematics?
Mathematics has been described as a form of knowledge which requires internal validity or
coherence. Does this make it self-correcting? What would this mean?
How is a mathematical proof or demonstration different from, or similar to, justifications
accepted in other Areas of Knowledge?
Is mathematical knowledge certain knowledge? Can we claim that '1 + 1 = 2' is true in
mathematics? Does '1 + 1 = 2' hold true in the natural world?
Does truth exist in mathematical knowledge? Could one argue that mathematical truth
corresponds to phenomena that we perceive in nature or that it coheres, that is, logically
connects, to a designed structure of definitions and axioms?
Over >>
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Mathematics and Knowledge Claims (continued)
Fermat's 'Last Theorem' remained unproved for 358 years, until 1995. Is mathematical
knowledge progressive? Has mathematical knowledge always grown? In this respect, how
does mathematics compare with other Areas of Knowledge (for example, history, the
natural sciences, ethics and the arts)? Could there ever be an 'end' to mathematics? In
other words, could we reach a point where everything important in a mathematical sense is
known? If so, what might be the consequences of this?
Has technology, for example, powerful computers and electronic calculators, influenced
the knowledge claims made in mathematics? Is any technological influence simply a
matter of speed and the quantity of data which can be processed?
What impact have major mathematical discoveries and inventions had on conceptions of
the world?
Mathematics and Values
Why do many mathematicians consider their work to be an art form? Does mathematics
exhibit an aesthetic quality?
What could be meant by G H Hardy's claim that: 'The mathematician's patterns, like the
painter's or poet's, must be beautiful; the ideas, like the colours or the words, must fit
together in a harmonious way. Beauty is the first test. There is no permanent place in the
world for ugly mathematics'?
What relationships, if any, exist between mathematics and various types of art (for
example, music, painting, and dance)? How can concepts such as proportion, pattern,
iteration, rhythm, harmony and coherence apply both in the arts and in mathematics?
Is the formation of mathematical knowledge independent of cultural influence? Is it
independent of the influence of politics, religion or gender?
What is meant by S Ramanujan's comment that 'Every time you write your student number
you are writing Arabic'?
If mathematics did not exist, what difference would it make?
Einstein, Albert (1879-1955)
The truth of a theory is in your mind, not in your eyes.
In H. Eves Mathematical Circles Squared, Boston: Prindle, Weber and Schmidt, 1972.
Euler, Leonhard (1707-1783)
Mathematicians have tried in vain to this day to discover some order in the sequence of prime numbers, and we have
reason to believe that it is a mystery into which the human mind will never penetrate.
In G. Simmons Calculus Gems, New York: McGraw Hill Inc., 1992.
Hardy, Godfrey H. (1877 - 1947)
I believe that mathematical reality lies outside us, that our function is to discover or observe it, and that the theorems
which we prove, and which we describe grandiloquently as our "creations," are simply the notes of our observations.
A Mathematician's Apology, London, Cambridge University Press, 1941.
Kepler, Johannes (1571-1630)
The chief aim of all investigations of the external world should be to discover the rational order and harmony which has
been imposed on it by God and which He revealed to us in the language of mathematics.
Kronecker, Leopold (1823 - 1891)
God made the integers, all else is the work of man.
(Die ganzen Zahlen hat Gott gemacht, alles andere ist Menschenwerk.)
Jahresberichte der Deutschen Mathematiker Vereinigung.
Sullivan, John William Navin (1886 - 1937)
The mathematician is entirely free, within the limits of his imagination, to construct what worlds he pleases. What he is
to imagine is a matter for his own caprice; he is not thereby discovering the fundamental principles of the universe nor
becoming acquainted with the ideas of God. If he can find, in experience, sets of entities which obey the same logical
scheme as his mathematical entities, then he has applied his mathematics to the external world; he has created a branch of
science.
Aspects of Science, 1925.
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TOK
TOK
Mathematics – Conjecturing & Proof
Circles, Points, Chords and Regions
Place n points on a circle and draw all the possible chords between points.
Do this in such a way so that three chords do not intersect at the same
point. This will insure that the number of regions formed is a maximum.
Make a conjecture for a pattern between the number of points n and the
number of regions r. Express your conjecture as a formula where r is equal to
an expression in terms of n. Do you think your conjecture is true for all cases?
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Theory of Knowledge
Question:
Is mathematics invented or discovered?
Some Replies / Ideas:
1. It is invented, just like any other language.
2. Mathematics is neither invented nor discovered, the principle of devising mathematical formulae
is invented, but the actual mathematics is just "there". Take Fibonacci numbers for example... Mr
Fibonacci sat down one day and wondered what would happen if you take 1, add 1 to it, then take
the result, and add that to the last number you came up with, then repeat for ever. Now if you
divide these numbers by each other, you get a number that decreases to the point where it gets
towards 1.618 (denoted by the Greek letter phi). This number ratio occurs naturally in many
situations, also known as the golden mean. It is the ratio of the sides of the rectangle most
pleasing to the eye. If a line segment is divided into two pieces (one shorter, one longer) such
that the ratio of the original segment to the longer piece is the same as the ratio of the longer to
the shorter piece, then this ratio is phi. Now is this discovered or invented?
3. Mathematics is a finite series of abstract two-dimensional symbols invented by humans, arranged
and rearranged according to a set of rules and procedures also invented by humans. When all the
procedures have been carried out and the rules adhered to an answer is produced. By this method
humans delude themselves into believing that they are revealing the secrets of the universe. If
you change the familiar symbols of mathematics to bones or tarot cards you have Magic and
witchcraft. It’s all the same. Incidentally, John Maynard Keynes (famous economist) once said of
Sir Isaac Newton (Mathematica Principia), that he was not so much the first of the great
scientists as the last of the great magicians.
4. Mathematics is both invention and discovery. The language of mathematics (such as addition &
fractions) is an invention. The things that the language describes (such as the Theorem of
Pythagoras) are discoveries. Both strands make up mathematics.
5. Math as a base ten system with Arabic numerals is invented. The basic relationships between
numbers are discovered, and described with math that is invented.
6. Just as the ability to formulate grammar seems to be embedded in genetic code, it's entirely
possible that a mathematical order is embedded in the universe, something that we are
discovering. We may not have perfected the tools (such as the decimal system, or binary code)
but no one can deny that certain parameters relate with others in a defined way. If mathematics
is a tool then we also 'discovered' that tool. I would suspect our earliest agricultural ancestors
discovered mathematics and I suspect the moon's waxing and waning would be the source of that
discovery.
7. Mathematics is the expression of the physical and metaphysical truths that surround us (and
always have). We have invented the language with which we describe the patterns we recognize
but we have not caused them to be true. Only the creator (whatever name you use for this entity)
has the power, cleverness and clearness of thought to create a physical law at will.
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Is Mathematics Invented or Discovered?
Ask a philosopher “What is philosophy?” or a historian “What is
history?”, and they will have no difficulty in giving an answer. Neither
of them, in fact, can pursue his own discipline without knowing what he
is searching for. But ask a mathematician “What is mathematics?” and
he may justifiably reply that he does not know the answer but that this
does not stop him from doing mathematics.
François Lassere
author of The Birth of Mathematics in the Age of Plato
Some questions to ponder:
What is mathematics? What are numbers and how did we come to discover them? But, did we
really discover them – perhaps we merely invented them? Is doing mathematics a natural activity
for the human mind or just a curious skill that is possessed by a few?
At the other end of the spectrum from these simple questions, there might be questions of a more
esoteric nature.
How did our ancestors pass from the mundane activity of counting to the concept of ‘number’ in
the abstract, separate from any particular collection of objects to label?
And did this progression launch us into an immaterial world of mathematical truths which we can
tune into because of some unusual propensity of the human mind?
If so, does this mean that mathematics is really a religion, albeit a rather austere and difficult
one? If we contacted some alien intelligence elsewhere in the universe, would it possess the
same type of mathematical knowledge that we do?
What is a priori knowledge? Is mathematical knowledge a priori knowledge?
What is a posteriori knowledge? Is mathematical knowledge a posteriori knowledge?
What is Platonism? And what does it have to do with ideas about absolute truth? If there is no
absolute (ideal) truth independent of human thought than what is an alternative
approach to the origin of human knowledge?
What does Euclid’s Fifth Postulate and non-Euclidean geometry have to do with
all of this talk about whether mathematical knowledge is discovered or invented?
Euclid
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Miscellaneous Recommendations/Suggestions
(notes from previous workshops)
External Assessment (80%)
o Formula (information) Booklet: students should know it well; encourage students to
have it available when working on all assignments and assessments (in or out of class)
o Exam format:
paper 1 (no GDC) and paper 2 (GDC allowed).
Section A (shorter problems) and section B (longer problems)
Assessment practice and design: tests and quizzes should model exam format
Specimen paper questions available on OCC
IB QuestionBank software is very helpful resource for making assignments
and assessments incorporating past exam questions and markschemes (be
aware that there quite a few typographical errors)
First exams 2006 – set as papers (new syllabus, but not the new exam
format – not separated into GDC/nonGDC questions)
First exams 2008 and onwards – follows GDC/nonGDC exam format
Should take place in January or February of Year 2.
Try using an old exam (but make sure it is new exam format)
o Display IB calculator policy poster in your classroom (IB coordinator can provide).
Internal Assessment (Portfolio) (20%)
o Good goal: one practice (early in year 1 – half way?); 4 portfolio – 2 type 1; 2 type 2.
This way students can choose their best two. (at least one, preferably 2 complete by
the end of the first year)
o 2 required forms for moderation sample: 5/IA (1 per group); 5/PFCS (1 per student).
Use Form B (from Teacher Support Manual – IA) to give feed back to
students w moderation sample instead of reverse side of PFCS form. More
useful to both student and moderator
Use tasks set by IB. Eventually try writing your own, but at first, use the ones
that already exist. You know they meet the requirements.
o Choose your favourites. Don’t give them your best one first – their first attempt will
likely not be their best.
o Technology Toolbox – show students what resources are available through the school.
Coordinate with science teachers
o About 2 weeks for tasks.
Two Q/A sessions (one near the beginning; one near the end)
Monitor students closely. Guard against plagiarism and collaboration.
o Try to respond to questions with other questions when talking with students. Guide
their work, don’t give them the solutions.
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Miscellaneous
o Graphic Display Calculator:
Teacher should have GDC same/similar to students; and a CAS (Computer
Algebra System) calculator. But if you show the CAS calculator to kids,
emphasize that they CANNOT use it for the exams.
Perhaps have school purchase same model of GDC and then sell to students.
This helps to ensure all students have the same model.
o Curriculum Review
Begin teaching ‘new’ syllabus in August 2012; first exams May 2014
Curriculum review documents exist on OCC
o Level Changes within the school
Moving from HL → SL; SL → Studies; Moving from SL→ HL or
Studies→ SL.
Coordinate as much as possible with department so courses overlap as much
as possible to make switching between courses easier (it’s obviously easier to
move “down” then “up”)
Recommend (don’t require) students to move early on in the year (though not
too early – make sure you have some good, solid assessments to base your
recommendation on)
o TOK – include in your class as you can; coordinate with the TOK teacher. Have fun
with it and make the students think.
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~ Diversion #6 ~
Diagonals of a Cube
What is the measure of the acute angle between two diagonals of a cube?
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