Workshop Resource Book

IB Math HL & SL workshop

IB Mathematics Higher Level &

IB Mathematics Standard Level

Workshop Resource Book

4-5 September 2012

Izmir, Turkey

Tim Garry

[email protected]


** 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)

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1 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

2 10:45-12:00 Overview of new internal assessment for HL & SL (Exploration). Management of the new IA – scheduling, ideas, resources, forms.

12:00-13:15 lunch

3 13:15-14:30 Samples of student Explorations. Marking a student Exploration. Teacher Support Material (TSM) for IA.


4 15:00-16:15 Technology in teaching and assessment. IB exams and GDC use. Resources for Math HL & SL – textbooks, software, websites.

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5 9:00-10:15 Theory of Knowledge activity. Math SL & HL mock exams – samples. Writing a mock exam. Use of past exams.


6 10:45-12:00 Management of current IA (Portfolio Tasks). Closer look at new syllabus content in SL & HL.

12:00-13:15 lunch

7 13:15-14:30 IB exam structure – Paper 1 & Paper 2. Use of exam markschemes. Marking student samples.


8 15:00-16:15 Sharing teaching ideas & materials. Preparing students for exams. Current and future developments in mathematics education.


~ Diversion #1 ~ pg.6






IB Math HL & SL Workshop Booklet

Table of Contents

IBDP & Group 5

IB Learner Profile 4

Aims & Objectives 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

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 118

Set of 13 SL Unit Tests 123

Theory of Knowledge (TOK)

Mathematics – TOK Questions 147

TOK Activity – Conjecturing & Proof 149

Is Mathematics Invented or Discovered? 150

Miscellaneous Recommendations / Suggestions 152


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 problem-solving

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


~ Diversion #1 ~


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)


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)


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Summary of Changes to Maths SL course

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.

2006 SL course 2014 SL course Changes


1. Algebra (8 hrs) 1. Algebra (9 hrs)

Calculation of binomial coefficient

using both GDC and formula.

2. Functions and

Equations (24 hrs)

2. Functions and

Equations (24 hrs)

Encouragement to connect with

applications in physics, chemistry

economics, etc.

3. Circular Functions &

Trigonometry (16 hrs)

3. Circular Functions &

Trigonometry (16 hrs)

Know exact values of trigonometric

ratios of 0, , , ,6 4 3 2

and their multiples.

4. Matrices (10 hrs) Content on matrices removed.

5. Vectors (16 hrs) 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.


Summary of Changes to Maths HL course


Summary of changes to HL options


Math HL

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

Syllabus content 140 hrs

Topic 1 - Algebra 8 hrs Topic 2 - Functions and equations 24 hrs Topic 3 - Circular functions and trig 16 hrs Topic 4 - Matrices 10 hrs Topic 5 - Vectors 16 hrs Topic 6 - Statistics and probability 30 hrs Topic 7 - Calculus 36 hrs

Portfolio 10 hrs

Total 150 hrs

Core syllabus content 190 hrs

Topic 1 - Algebra 20 hrs Topic 2 - Functions and equations 26 hrs Topic 3 - Circular functions and trig 22 hrs Topic 4 - Matrices 12 hrs Topic 5 - Vectors 22 hrs Topic 6 - Statistics and probability 40 hrs Topic 7 - Calculus 48 hrs

Portfolio 10 hrs

Option syllabus content 40 hrs

Total 240 hrs

Math SL

Topic 1 - Algebra


Math HL

Topic 2 - Functions and Equations

Math SL


Math HL

Math SL


Math HL

Topic 2 - Circular Functions and Trigonometry

Math SL


Math HL

Topic 4 - Matrices

Math SL


Math HL

Topic 5 - Vectors

Math SL


Math HL

Topic 6 - Statistics and Probability


Math SL Math HL


Math HL

Topic 7 - Calculus

Math SL


Math HL

Math SL


Math HL

credit: Wiley Miller, Universal Press Syndicate permission for classroom use only


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, 10na ,

1 10a , 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


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 1 2m m and

1 2 1m m

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

1.3 The binomial theorem: expansion of ,n

a b n ; calculation of binomial

coefficients using Pascal’s triangle and the formula for n

r

, also written as n rC

2. Functions and Equations

2.1 Concept of function :f x f x ; domain, range; composite functions; identity

function; inverse function 1f

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 1y f 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

the x-direction with scale factor 1

b: y f bx ; composite transformations

2.4 The quadratic function 2x ax bx c : its graph, y-intercept 0,c ; axis of symmetry;

‘factored’ form: x a x p x q , x-intercepts ,0p and 0, p ; ‘vertex’ form:

2

x a x h k , vertex ,h k

2.5 The reciprocal function 1

, 0x xx

; its graph and self inverse nature; the rational

function ax b

xcx d

and its graph; vertical and horizontal asymptotes


2.6 Exponential functions and their graphs: , 0xx b b ; and xx e ; logarithmic

functions and their graphs: , 0xx b b , xx e ; logarithmic functions and their

graphs: log , 0bx x x , ln , 0x x x ; relationships between these functions:

lnx x bb e ; log x

b b x ; log, 0b x

b x x

2.7 Solving equations, both graphically and analytically. Use of technology to solve a variety

of equations. Solving 2 0, 0ax bx c a ; the quadratic formula; the discrminant 2 4b ac 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 sin

cos

;

exact values of trigonometric ratios of 0, , , ,6 4 3 2

and their multiples

3.3 The Pythagorean identity 2 2sin cos 1 ; double angle identities for sine and cosine;

relationship between trigonometric ratios

3.4 The circular functions sin , cos and tanx x x ; their domains and ranges; amplitude, their

periodic nature; and their graphs; composite functions of the form

sinf x a b x c d ; transformations

3.5 Solving trigonometric equations in a finite interval, both graphically and analytically;

equations leading to quadratic equations in sin , cos or tanx x x

3.6 Solution of triangles; the cosine rule; the sine rule, including the ambiguous case; area of

a triangle 12

sinab C

4. Vectors 4.1 Vectors as displacements in the plane and in three dimensions; components of a vector;

column representation;

1

2 1 2 3

3

v

v v v i v j v k

v

; algebraic and geometric approaches to

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; , and i j 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


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

probability of an event A is

P

n AA

n U ; the complementary events A and A (not A);

use of Venn diagrams, tree diagrams and tables of outcomes

5.6 Combined events, P A B ; mutually exclusive events, P 0A B ; conditional

probability; the definition

PP

P

A BA B

B

; independent events; the definition

P P PA B A 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

principles as

0limh

f x h f xf x

h

; derivative interpreted as gradient function

and as rate of change; tangents and normals, and their equations

6.2 Derivative of nx n , sin x , cos x , tan x , xe 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 , and f f f ; optimization

6.4 Indefinite integration as anti-differentiation; indefinite integral of nx n , sin x ,

cos x , 1

x and xe ; the composites of any of these with the linear function ax b ;

integration by inspection, or substitution of the form f g x g x dx

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


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


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.


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 (pre-May 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.


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)


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, 3112 u and 205 S .

(a) Find

(i) the common difference;

(ii) the first term. [4 marks]

(b) Find 10S . [3 marks]

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The quadratic function f is defined by 32)( 2 xxxf .

(a) Write f in the form khxxf 2)()( . [2 marks]

(b) On the grid below, sketch the graph of f clearly marking any important points.

[4 marks]

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(a) Given that 3sin3cos2 2 find the two values for sin . [4 marks]

(b) Given that 3600 and that one solution for is 30 , find the other two

possible values for . [2 marks]

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(a) (i) Factorise the equation 032 2 uu .

(ii) Hence, or otherwise, solve the equation 032)2(2 2 xx .

[4 marks]

(b) Solve 2121log x [2 marks]

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Show that xxx

xx

dx

d

2sin1

2

sincos

sincos

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6. [Maximum Mark: 6]

(a) Find dxx 3

6 [3 marks]

(b) Find 3

0)3( dxxx [3 marks]

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A coin is biased so that P(Head) 3

2 and P(Tail)

3

1

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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Section B

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


(a) Write out the sixth line of Pascal’s Triangle [1 mark]

(b) Consider the expansion of 5)1( x

(i) Write down the first four terms of this expansion

(ii) Use your answer to (b)(i) to evaluate 5)003.1( , giving your answer

correct to 7 decimal places. [6 marks]

(c) Find the term independent of x in the expansion of

15

2 32

xx , leaving

your answer in the form qp

r

n baC where qpba &,, are integers. [5 marks]


Consider the function 2

3)(

x

xxf .

(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 )(xf showing all relevant detail. [3 marks]



(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 %62.0 of these

batteries last longer than 50 hours, find the standard deviation. [5 marks]


Consider the function 13: 2 xxf

(a) Find the area enclosed by the curve, the lines 1x and 2x , 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]


sample mock Paper 1 exam for Mathematics Standard Level

MARKSCHEME Section A

1. (a)

(i)

]11[3

279

}2{}1{

}2{4220)42(2

5

}1{3111

11

1

AMd

d

dudu

du

(ii) ]11[23133 11 AMuu

(b) ]12[115))3(9)2(2(510 AMS

2. (a)

]11[4)1(

31)12(

32

2

2

2

AMx

xx

xx

(b) Graph is a parabola passing through:

(-3,0), (-1,4), (0,-3) and (1,0) [3A]

Overall shape [1A]

3. (a)

]22[12

1sin

0)1)(sin1sin2(

03sin3sin2

03sin3sin1(2

3sin3cos2

2

2

2

AMor

(b) ]2[90150 Aor

4. (a)

(i) ]1[0)1)(32( Auu


(ii)

]11[012

]1[2

32

12

32

12

3

2

AMx

Asolutionno

or

oru

uLet

x

x

x

x

(b) ]11[111212 AMxx

5.

]7[2sin1

2

cossin21

2

cossin2sincos

cossin2sincoscossin2sincos

)sin(cos

)cossin)(sin(cos)cossin)(sin(cos

cossinsincos

cossinsincos

22

2222

2

Mx

xx

xxxx

xxxxxxxx

xx

xxxxxxxx

dx

dy

xxdx

dvxxv

xxdx

duxxu

6.

(a) ]12[3

2

66

2

23 AMc

xc

xdxx

(b)

2

9

2

233

3

)3(

3

0

3

0

2

xx

dxxx

However, the negative simply refers to the position of the area in relation to the x-axis,

Hence required area is ]12[2

9 2 AMu


7.

(a)

]11[243

80

3

1

3

2)4(

04

4

5 AMCXP

(b) Zero tails 5 heads = ]11[243

32AM

(c) <3 heads implies 0, 1 or 2 heads

]12[243

51

243

40

243

10

243

1AM

Section B

8.

(a) 1 5 10 10 5 1 [1A]

(b)

(i)

]11[101051 32 AMxxx

(ii)

]11[)7(0150903.1

01509027.1

00000027.000009.015.01

]11[)000000027.0(10)000009.0(10)003.0(51

AMdp

AM

(c)

]23[32is termrequired hence

100330 if oft independen is termthe

323

2

105

10

15

330151515215

AMC

rrx

xCx

xC rrr

r

rr

r

9. (a) ]1[2 Ax

(b) ]2[1 Ay

(c) ]2[2

3Ay

(d) ]2[3 Ax


(e)

]12[points turningno

0for which of valueno is There

)2(

52

RM

dx

dyx

xdx

dy

(f)

10. (a)

(i)

]13[3085.0

6915.01

)5.0(1

)5.0()40(

2

1

10

4540 is units standardin 40

)40(

)10,45(~ 2

AM

ZPXP

XP

XZ

NX

(ii) AM 1130853085.010000

2

3

1

2 3

[3A]


(b)

]14[2

5.24550

5.2

9938.0)(

0062.0)(1

0062.04550

0062.0)50(

),45(~ 2

AM

a

a

a

ZP

XP

NX

11 (a)

]12[8

210

13

2

2

1

3

2

1

2

AMu

xx

dxx

(b)

]15[5

354

125

92)8(2)32(

5

9

3

6

5

9

169

13

2

1

35

2

1

24

2

1

22

AM

xxx

dxxx

dxx


… MOCK EXAM ……… MOCK EXAM ……… MOCK EXAM …

MATHEMATICS

HIGHER LEVEL Name

PAPER 1 Friday 11 March 2011 Candidate session number

2 hours 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.


0 0


– 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.


When the function 4 3 26 11 22 6f x x x x ax is divided by 1x the remainder is 20 .

Find the value of a.

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– 3 –


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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– 4 –


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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– 5 –


Find the coordinates of the point where the line with the vector equation

4 2

2 1

2 3

r

intersects the plane with the equation 2 3 2x y z .

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– 6 –


(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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– 7 –


Find the equation of the line that is tangent to the curve 2 23 4 7x y where 1x and 0y .

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– 8 –


Find the value of x satisfying the equation

2 1 23 4 6x x x

Give your answer in the form ln

ln

a

b where ,a b .

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– 9 –


The independent events A and B are such that P 0.4A and P 0.88A B . 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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– 10 –


The area of the enclosed region shown in the diagram is defined by

2 2, 2, where 0y x y ax a

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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x

y

2

a0


– 11 –


The diagram below shows the graph of equation 1 , 0 4y f x x .

On the axes below, sketch the graph of 20

x

y f t dt , marking clearly the points of inflexion.


– 12 –

Section B



The probability density function of the random variable X is given by

2, for 0 1

4

0, otherwise

kx

f x x

(a) Find the value of the constant k. [5 marks]

(b) Show that 6 2 3

E X

[7 marks]

(c) Find the median of X. [5 marks]


(a) Find the root of the equation 2 2 2x xe e giving the answer as a logarithm. [4 marks]

(b) The curve 2 2 2x xy e e has a minimum point. Find the coordinates of

this minimum. [7 marks]

(c) The curve 2 2 2x xy e e is shown below.

Write down the coordinates of the points A, B and C. [3 marks]

(d) Hence state the set of values of k for which the equation 2 2 2x xe e k

has two distinct roots. [2 marks]


– 13 –


(a) Show that the following system of equations will have a unique solution

when 1a .

2

3 0

3 5 0

5 2 9

x y z

x y z

x y a z a

[5 marks]

(b) Given that 1a , state the solution in terms of a. [6 marks]

(c) Hence, solve

3 0

3 5 0

5 8

x y z

x y z

x y z

[2 marks]


(i) Using mathematical induction, prove that

1

1

1 2 2n

r n

r

r n

[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]



MATHEMATICS HIGHER LEVEL PAPER 1

► MARKSCHEME ◄

1.

2.

[5 marks]

(C5) [5 marks]


3.

4.

5.

(C6) [6 marks]

(M1)

(A1)

(A1) (C3)


6.

Finding equation of line; slope is 3

4 and passes through 1,1 (M1)

3

1 14

y x (M1)

3 7

4 4y x (A1) [7 marks]

7.

[6 marks]

(A2)

(A2)


8.

9.

10.

[6 marks]

[6 marks]

(M1)

(A1)

[6 marks]


11.

[5 marks]

Total [17 marks]


12.


13.


14.

continued on next page …

[4 marks]


14. (continued)

Total [14 marks]

[4 marks]

[4 marks]



MATHEMATICS

HIGHER LEVEL Name

PAPER 2 Friday 18 March 2011 Candidate session number

2 hours 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.


0 0


– 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.


Triangle ABC has C 42 , BC 1.74 cm, and area 1.19 2cm .

(a) Find AC. [2 marks]

(b) Find AB. [4 marks]

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– 3 –


Find the values of a and b, where a and b are real, given that 2 5a bi i i

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– 4 –


The function f is defined as 3 4

, 22

xf x x

x

.

(c) Find an expression for 1f x . [5 marks]

(d) Write down the domain of 1f [1 marks]

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– 5 –


The function f is defined as sin ln for 0.5,3.5f x x x x .

(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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– 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 –


Find 2 sinxe x dx .

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– 8 –


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 –


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 –


Solve the differential equation

2

2 4 2dy

x xy xdx

given that 1 when 1y x .

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– 11 –


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 2cm sec . When 2x , 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



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

2, for 0

1

0 otherwise

xx k

f x x

(a) Find the exact value of k. [5 marks]

(b) Find the mode of X. [2 marks]

(c) Calculate P 1 2X . [3 marks]


– 13 –

13. [Total mark: 26]

Part A [Maximum mark: 14]

(b) The line 1L passes through the point A 0,1, 2 and is perpendicular to the plane

4 3 0x y z . Find a Cartesian equation of 1L . [2 marks]

(b) The line 2L is parallel to

1L and passes through the point 3, 8, 11P . Find the

vector equation of the line 2L [2 marks]

(c) (i) The point Q is on the line 1L such that PQ is perpendicular to 1L and 2L .

Find the coordinates of Q.

(ii) Hence find the distance between 1L and 2L . [10 marks]

Part B [Maximum mark: 12]

Consider the system of equations

2 0

3 3

8 5 6

x y kz

x y z

kx y z

(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]


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.


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 2x on your GDC, do NOT write

nDeriv(Y£,X,3)=6. You should write 3 6f or 6dy

dx when 3x . Other examples

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 , 906 4 2

and multiples of these values.

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 sin90 .

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 !


~ Diversion #2 ~

Two algebraic puzzles

► Proof that 2 1 ◄

Let a b

Multiply both sides by a: 2a ab

Subtract 2b from both sides by a: 2 2 2a b ab b

Factorise: a b a b b a b

Divide both sides by a b : a b b

But a b ; therefore 2b b

Divide both sides by b: 2 1

► Proof that 1 0 ◄

2 21 2 1x x x

2 21 2 1x x x

2 21 2 1 2 1 2 1x x x x x x x

2 2 221 1

4 41 1 2 1 2 1 2 1 2 1x x x x x x x x

2 2

1 12 2

1 2 1 2 1x x x x

1 12 2

1 2 1 2 1x x x x

1x x

1 0


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


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



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


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



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



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.



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 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


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


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.



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



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.


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


1 There is some appropriate mathematical presentation.

2 The mathematical presentation is mostly appropriate.

3 The mathematical presentation is appropriate throughout.

C Personal Engagement


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


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


1 Some relevant mathematics is used.

2 Some relevant mathematics is used. Limited understanding is demonstrated.

3 Relevant mathematics commensurate with the level of the course is used. Limited

understanding is demonstrated.

4

Relevant mathematics commensurate with the level of the course is used. The

mathematics explored is partially correct. Some knowledge and understanding are

demonstrated.

5


mathematics explored is mostly correct. Good knowledge and understanding are

demonstrated.

6


mathematics explored is correct. Thorough knowledge and understanding are

demonstrated.


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? Yes No

(preferably no full-page graphs; and no separate appendices)

■ 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

( 2y x , not ^ 2y x ; , 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


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


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.


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 whole-class 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.


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


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.


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? Yes No

(including handwritten labels for calculator screen images)

■ 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? Yes No

(preferably no full-page graphs; and no appendix containing graphs)

■ 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

( 2y x , not ^ 2y x ; , 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? Yes No (“other situations” can include a change of parameter or more data)

■ 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: ________________________________________ Student: ____________________________


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


Mathematics SL: The Portfolio Form B

Feedback to student Form B

Name:

Title of task: Type: I II

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: / 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?


Algebra Preparation for Math SL/HL

25 Exercises (worked solutions on following pgs)

1. Expand: 2

a b

2. Simplify: 5 4

3 6

x

3. Simplify: 1

6 8

k

k k

4. Solve for x: 1

6 8

x

x x

5. Expand: 3

2 2m

6. Simplify:

2 24 4x h x

h

7. Simplify:

a a

b c

c

8. Simplify: 2

xx

y

xx

y

9. Factorise: 3 23 5 12p p p

10. Factorise: 225 40 16m m

11. Simplify:

2

3

5 1 3 2 1

1

x x x

x

12. Simplify: 1 1

a b a b

13. Simplify: 2

2

4 3 4 1

4

x x x

x

14. solve for k: 2

7 3mk

15. solve for y: 42

xx

y

16. Solve for b: 6a

b c b

17. Solve for x: 5 7x

18. Solve for x: 5 7x x

19. Solve for w: 4 23 4 0w w

20. Solve for x: 5 9x

21. Simply:

52

2

2

xx

x

22. Solve for x: 2 5 5x x

23. Solve form m: 22 7 4m m

24. Solve for x: 3

12

x

x

25. Is there anything incorrect with the

following solution ?

2 6 4 3t t t

2 6 4 3 3 2 4 3t t t t t t

3t 2

3

t

t

4 3t

3t 2 4 2t t

therefore, the solution is 2t


Worked Solutions – Algebra Preparation Exercises for Math SL/HL

1. 2 222a b ba b a

2. 10 45 4 2 5 4

3 6 2 3 6 6

14

6

xx x x

3. 1 1 8 6

6 8 6 8 8 6

k k k k

k k k k k k

2

2

2

22

8 6

2 48 2

7 8

28 484

k k k

k k k k

k k

k k

the denominator factorises (numerator does not), but writing answer as

2 7 8

6 8

k k

k k

is not

considered simplifying (only factorise for a reason); best to leave simplified answer as 2

2

7 8

2 48

k k

k k

4. 216 8 7 8 0

6 8

xx x x x x

x x

could solve this equation with quadratic formula, but since the quadratic expression 2 7 8x x

factorises then solving by factoring is preferred since it is more efficient (and less prone to errors)

2 7 8 0 8 1 0x x x x

8 0 OR 1 8 OR 0 1xx x x

5. 3

2 2 4 22 2 4 4m m m m

6 4 2 64 2 4 24 4 2 8 6 128 8m m m m m m m m

6. 2 2 2 22 2 2 2 24 2 44 4 4 8 4 4 8 4x xh h xx h x x xh h x xh h

h h h h

8 48 4h x h

hx h


7. 2

1

1 1

a a ac ab ac ab

ac abb c bc bc bcc cc bc c

ac ab

bc

8.

2 2

2 2 2 2 2 2 2 2

2 2 2

1

x x xy x xyx

x yx xy y yy y y y

xy x xy x y xy x y x y xxx

y y yy

2

22

1y y

y x

y y

x y

9. 3 2 23 5 12 3 5 1 3 4 32 p p pp p p p p p

10. 2 25 425 40 16 5 4 5 4m m m m m

11.

2

3 3

1 5 1 3 25 1 3 2 1

1 1

x x xx x x

x x

2 2

5 5 3 6

1

2 11

1

xx

x x

x

12. 2 2

1 1 2a b a b

a b a b a b a b a b a b

b

a b

13. 2 2 2 2

2 2 2

4 3 4 1 4 3 4 4 4 4

4 4 4

x x x x x x x x

x x x

2

2 2

2

2

2x x

x x

x

x

14. 2

7 3 7 3 22

7 3m k km

mk


15. 4 2 4 4 2 82

xx x y x x xy y x

y

3 8 4 3 88

43

4x xy y x y x

xy

x

16. 6

6 6 6 6 6a

b c ab b c ab b ab cb c b

6 6

6 OR 6 6

6c c

b ba

b a ca

17. 5 7 5 49 44x x x

18. 2 2 2 25 7 5 49 14 0 15 44 4 11 0x x x x x x x x x

4 OR 11x x but need to check answers (why?)

checking answers 4x : 4 5 7 4 9 3 OK

11x : 11 5 7 11 16 4 NO

thus, the only solution is 4x

19. incorrect solution:

23

6 4 3 3 2 4 3t

t t t t t t

2

3

t

t

4 3t

3t 2 4 2t t

therefore, the solution is 2t

correct solution:

2 2 26 4 3 6 4 12 5 6 0 2 3 0t t t t t t t t t t

2 OR 3t t

by dividing both sides of equation by 3t the solution of 3t was ‘lost’ in incorrect solution

20. let 2w t : 4 2 23 4 0 3 4 0 4 3 0 4 OR 3w w t t t t t t

substitute t back in for 2w : 2 4 2w w OR 2 3w no real solution

therefore, there are two solutions 2 OR 2w w


-412

2m – 1

m + 4 +

+

– 0

0

sign chart

m

– –

+

(2m – 1)(m + 4) + +–

solution: 1

4 or 2

m m

21. 5 9 5 9 OR 5 9x x x

14 OR 4x x

22.

5 2 5 32

3 12 2 2 222 2 22

1

x xx

xx x x xxx x xx

1 3223

3 1 3 3 OR

22

2 2

x x

x

x

xx x

23. 2

22 2 25 5 5 5 5 5x x x x x x

2 25 25 1 20 10 20x x x xx

24. 2 22 7 4 2 7 4 0 2 1 4 0m m m m m m

25. 3 23 3 3 2

1 1 0 0 02 2 2 2 2

x xx x x x

x x x x x

5

0 22 02

xx

x


► Set of 13 Unit Tests for Maths SL ◄

Unit 1 Test – Fundamentals

1. Simplify each radical expression.

(a) 10

8 (b) 12 27

2. (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. Simplify:

32

3 5

3 2

2

x y

x y

4. Simplify: 3 3

4 3

y y

5. Completely factorize the expression: 23 27x y y

6. Solve the system of equations using any method you wish.

4 6

8 3 13

x y

x y

7. Solve for x: 9

2 14

xx

x

8. Solve for k: 8 2 3 1m k m k

9. Simplify: 3 4

2 4

m

m m

10. Expand: 3

2 1a

11. Solve for w in terms of n in the formula: 2

1 53

wn w

n

Bonus Questions

(1) Rationalize the denominator 6

2 7 . Give answer completely simplified.

(2) Find exact coordinates of the points where line 1

2y x and circle 2 2 1x y intersect.


Unit 2 Test – Functions and Equations Math SL

1. Consider the following equation for a parabola 2 8 11y x x

(a) Write the equation in ‘vertex form’, that is in the form 2

y a x h k . .

(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 2

2f x

x

(a) State the domain and range for f x

(b) Find 1f x .

(c) State the domain and range for 1f x

3. State the domain and range for the function.

2

1

9f x

x

4. Solve the following quadratic equation by the method of ‘completing the square’.

22 12 5 0x x

5. Given 3

8f xx

and 1

1g x

x

, find the following – completely simplified.

(a) f g x (b) g f x (c) 1

f g x

6. The graph at right shows y g x

On the coordinate planes provided on the next two pages, sketch the following:

(a) 1g x (b) 3g x (c) 2g x (d) g x


6. (a)

(b)


6. (continued)

(c)

(d)


7. Consider the function 2 3 4f x x

(a) Sketch a graph of the function on the grid below.

(b) Find any x- or y-intercepts.

(c) State the domain and range.

(d) Does f have an inverse function, i.e. 1f x ? 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.

(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 5y g x ?

x- 1 0 - 5 5 1 0

y

- 1 0

- 5

5

1 0

(5, -1)

(-1, -13)


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 5

2x y - 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.

(a) 10

4

3 5i

i

(b)

17

1

24

3

r

r

[do not need to find sum; just first three terms]

7. Find the coefficient of 3x in the expansion of

5

3 2x

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. Find the term independent of x in the expansion

9

2 32x

x

.

Bonus Question

The consecutive terms of a geometric sequence are given by 2, 6 and 2 3x x x .

(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.


x- 2 2 4 6 8 10

y

- 1

1

( 9 , 1 )

( 0 , 0 )

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

(b) 3

532

2. Solve for x in each equation.

(a) 1

227 9xx

(b) log 16 2x (c) 8log 4 x

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) 4log 2 log 10b b (b) 2log 3log loga b c

4. Approximate the value of the logarithm to three significant figures. 3log 15

5. Solve for x: log

3log 10

b

b

x

6. State the domain and range of each function.

(a) ln 1f x x (b) 2 1xg x

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.

8. Find the equation that is shown in the graph.

9. Solve for x. Approximate answers to three significant figures.

(a) 2 1 40xe (b) 5log 2 10x (c) 3

4.17 2.36 274.9x


x- 5 - 4 - 3 - 2 - 1 1 2 3 4 5 6 7

y

- 5

- 4

- 3

- 2

- 1

1

2

3

4

5

6

7

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

initial amount 0A is given by 0

5001

2

t

A t A

where t is the number of years.

(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 xy e ?

(b) Sketch the graph of 1xf x e on the grid below.

(c) The graph of 1xy e has one asymptote. What is the equation of the asymptote?

(c) Sketch a graph of the inverse of 1xf x e . Label it 1f x . The graph of 1f x

has one asymptote. What is the equation of the asymptote?

(d) Find the equation for 1f x .

12. Solve for x. Give your answer exactly. 3 3log 1 log 9 2x x

Bonus questions:

(1) Find b: ln3x xe b

(2) Solve for x: 2

8log log 2

3xx


x-4

4

2

34

54

32

74

2 94

y

-3

-2

-1

1

Unit 6 Test – Trigonometric Functions and Equations Math SL

1. Write down the domain, range, period and amplitude for each function.

(a) 3cos 22

y x

(b) 2 sin4

xy

2. (a) Write the expression cos2 sinx x in terms of sin x only.

(b) Solve the equation cos2 sin 0x x for the interval 0 2x ; give answer(s) exactly.

3. The graph of a sine function siny a 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.

4. Find all solutions for 1

cos32

x in the interval 0 x . Give exact answer(s).

5. Given that 5

tan12

and that 02

, find the exact values of:

(a) sin (b) cos (c) sin 2

6. The average monthly precipitation for Aberdeen is modelled by the trigonometric function

15sin 3 656

P M

, P is precipitation in mm, and M is the month (Jan 1 , Feb 2, )

(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 3

sin 44

x have in the interval 0 x ?

Provide a very brief justification for your answer.

(b) Give the solutions for 3

sin 44

x , 0 x , approximately to 3 significant figures.


8. Solve the equation 22sin cos 1 0x x , 0 2x . 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

2cosh a b t

k

, 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

10. The diagram below shows the graph of 1 tan2

xf x

for 360 360x .

(a) Write down …

(i) the period of the function;

(ii) the value of 90f

(b) Solve 0f x for 360 360x


10 cm

10

cm

12 cm

12

cm

U

R

T

S

P

12 cm

12 cm

1.5

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 2

sin3

. 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.

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 .

(2) Find the angle measure between two diagonals of a cube.

k + 4

k + 6k + 2


O

B

A

C

a

b

Unit 8 Test – Vectors (can be given as two separate tests) Math SL

Part I

1. The position vectors of the points A and B with respect to the origin O are 2 3 and 5 i j i j

respectively.

(a) Find the position vector of the point C such that 2AC AB .

(b) Calculate the angle between the vectors and AB OB .

2. Find a vector of magnitude 8 units and is parallel to the vector 2i + 5j .

3. The angle between the vector 2 a i j and the vector 2 m b i 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 , 2, 3B and 2, 3C .

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

6. The points P, Q, R and S have position vectors 2 3 i j , 3 8i j , 7 6i j and 7 ni j

respectively, where n is a constant. Find the value of n such that PR is perpendicular to QS .

7. Given that 4

3

a ,

2

4

b and 22

11

c , find:

(a) a unit vector perpendicular to a;

(b) the value of the constants m and n such that m n a b c

(c) the measure of the angle between a vector in the direction of a and a vector in the direction

of b.


Unit 8 Test – Vectors (continued)

Part II

1. If

3

2

4

a and

4

2

1

b find c such that 4 2 a c b .

2. Find the angle between the vectors p and q given that 3 4 p i j and 5 12 q i 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 4 5 i j k and m n 15i j k 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

6. For

3

2

4

a

,

1

5

3

b

and

1

0

5

c

, find the following:

(a) 3a b c (b) b c (c) a unit vector parallel to b

(d) a b (e) a b b

7. HJKL is a rectangle and M is the midpoint of JK .

Express each of the following vectors in terms of MH and ML .

(a) LH (b) MJ (c) JH

8. Points A, B and C have position vectors 3 4i j , 3i j and 7 7i 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 .

H

J

L

KM


Unit 9 Test – Differential Calculus (two parts – given as two separate tests) Math SL

Part I: Fundamentals

1. Find the derivative, dy

dx, of each function.

(a) 3 27y x (b) 5

6y

x (c) 4y x

2. The line 16 9y x is a tangent to the curve 3 22 9y x ax bx at the point 1,7 . Find

the values of a and b.

3. The curve 3 22 4 4 3y cx x x has a stationary point at 1x . Determine the value of c.

4. Find the equation of the line tangent to the curve 3 5 2y x x where 1x .

5. Given that the function f is defined as 3 23 24 5f x x x x , (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 5 3

2

2 7x xy

x

, find

dy

dx. [hint: simplify

5 3

2

2 7x x

x

before differentiating]

7. Find the gradient of the tangent to the curve 3 5 1y x x at the point where 2x .

8. Find the equation of the tangent to the curve 22 3y x x at the point 1

, 22

.

9. Find the interval(s) in which the function 23 12 5f x x x is decreasing.

10. Consider the function 4 3 25 6 4h x x x x x

(a) Show that the graph of h has a horizontal tangent at 1

4x and at 2x .

(b) Determine whether h has a maximum, minimum or neither at 1

4x and at 2x .

(c) The graph of h has two inflexion points. Find the coordinates of one of these inflexion

points such that 1.x

11. Consider the function 3 27 5g x x x x with a domain of 0x .

(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 0x ?

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 3 29 27 , 0x t t t t t .

(a) When is the object stationary?

(b) What is the rate of change of the object (give appropriate units) at 1t second?

(c) Determine when the object’s acceleration is negative, and when it is positive.


2x

x

h

Unit 9 Test - Differential Calculus (continued) Math SL

Part II: Further Techniques & Applications

1. Given 21y x , find dy

dx. Simplify the expression for

dy

dx.

2. Consider the function sinxf x e x .

(a) Find the derivative of f with respect to x, i.e. find f x .

(b) On the interval 0 4x , find the exact value of x where 0f x .

(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 cos 2y x where 4

x

.

4. Given the function 2 ln , 0y x x x , find dy

dx and

2

2

d y

dx and simplify the expressions for each.

5. Find the equation of the line tangent to ln at the point where y x x x e

6. Explain why the curve 9

3y xx

has no stationary points.

7. Find the line normal to the graph of 2cos2

xy

where x . Express the line exactly.

8. Find the exact coordinates of the inflection point on the curve ln 2y x x .

9. Find the maximum area of a rectangle inscribed in an isosceles right triangle whose hypotenuse

is 20 cm long.

10. A six-sided box with a total surface area of 500 2cm is

constructed such that the length of its rectangular base is

twice as long as its width.

(a) If the width of the rectangular base is represented by x and the width by 2x , show that the

height, h, of the box can be expressed in terms of x as 2250 2

3

xh

x

.

(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.


x

y

(0,0) A

y = x ln x – x

Unit 10 Test – Integral Calculus Math SL

1. Find each of the following:

(a) 3 2 x dx (b) 3

6 1 dxx

2. By using a suitable substitution, find each of the following:

(a) 21

xdx

x (b) 2sin x x dx

3. An object moves in a straight line. At time t seconds the object’s velocity is given by

sinv t t t . When 0t 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 0 to 2t t . Give exact answer.

(c) What is the total distance traveled by the object from 0 to 2t t . Give exact answer.

4. Find the area of the region enclosed by the curves ln , xy x y e and the vertical line 4x .

5. (a) If 4 y x x , find dy

dx and simplify.

(b) Hence, evaluate 2

0

8 3

4

xdx

x.

6. Find the area of the region bounded by the graphs of 2 2 3 y x x and 1 y x .

7. Consider the function ln , 0 f x x x x x .

(a) The function has an x-intercept at the point A. Find

the exact x-coordinate of 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

region bounded by 6

yx

, 2x and 4x about the x-axis.


Unit 11 Test – Statistics Math SL

1. After an exercise class, a group of 24 people measured their heart rates. The data is below.

(a) Find the following:

(i) the lower quartile (ii) the median (iii) the upper quartile

(iv) the interquartile range (v) the range of the data

2. The cumulative frequency graph below shows the heights of 120 girls in a school.

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. (continued on next page)

125 118 144 133 123 120 140 148

128 112 108 98 156 144 126 135

108 100 96 105 112 148 106 85

(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.


4. 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.

5. The following is the cumulative frequency diagram for the heights of 30 plants given in

centimetres.

(a) Use the diagram to estimate the median height.

(b) Complete the following frequency table.

Height (h) Frequency

0 h 5 4

5 h 10 9

10 h 15

15 h 20

20 h 25

(c) Hence estimate the mean height.

6. 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)

cum

ula

tiv

e fr

equ

ency

height ( )h

30

25

20

15

10

5

5 10 15 20 25


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.

(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?

Hours 20 16 21 10 15 19 13 11

Exam score 95 90 85 83 77 62 58 49


Unit 12 Test – Probability Math SL

1 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.

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.

2 A coin is biased (i.e. not fair) so that 2 1

and 3 3

P head P tail . If the coin is tossed six

times find the probability of obtaining

(a) exactly five heads; (b) at least two heads; (c) at least one tail.

3 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; (b) two odd and two even numbers.

4. Given A and B are independent events, and that 0.64, 0.73, find p A p A B 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; (b) exactly one of them is rotten?

6. 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.


B V

11 8 4

2


7. Louis and Pierre play a match consisting of five games, each of which must be won or lost. In

each of the first three games the probability that Louis will win is 2

3 and in the remaining two

games the probability is 3

4. Find the probability that Louis will win four or more of the games.

8. If 0.8, 0.6P X P Y 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


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 0 1 2 3

P X x 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.

, 1, 2, 3, 4

0, otherwise

kx

P X x x

(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.


~ 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.


~ 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?


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?


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.


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


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.



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.


~ Diversion #6 ~

Diagonals of a Cube

What is the measure of the acute angle between two diagonals of a cube?

Workshop Resource Book

Documents

Transcript of Workshop Resource Book