# Mathematics: AA-HL Handbook

## Aims

## Analysis and Approaches Aims

The aims of DP mathematics courses are to enable students to:

1. develop a curiosity and enjoyment of mathematics, and appreciate its elegance and power

2. develop an understanding of the concepts, principles and nature of mathematics

3. communicate mathematics clearly, concisely and confidently in a variety of contexts

4. develop logical and creative thinking, and patience and persistence in problem solving to instill confidence in using mathematics

5. employ and refine their powers of abstraction and generalization

6. take action to apply and transfer skills to alternative situations, to other areas of knowledge and to future developments in their local and global communities

7. appreciate how developments in technology and mathematics influence each other

8. appreciate the moral, social and ethical questions arising from the work of mathematicians and the applications of mathematics

9. appreciate the universality of mathematics and its multicultural, international and historical perspectives

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

11. develop the ability to reflect critically upon their own work and the work of others

12. independently and collaboratively extend their understanding of mathematics

## Assessment objectives

By the end of the course, students are expected to reach the following assessment objectives.

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 abstract and real-world 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; use appropriate notation and terminology.

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 from the real world, involving organizing and analyzing information, making conjectures, drawing conclusions, and testing their validity

## Assessment objectives in practice

## Syllabus Outline

## Syllabus Content

## External Assessment Outline -HL

## External Assessment Details-HL

General information

Papers 1, 2 and 3

These papers are externally set and externally marked. Together, they contribute 80% of the nal mark

for the course. These papers are designed to allow students to demonstrate what they know and what

they can do.

Papers 1 and 2 will contain some questions, or parts of questions, which are common with SL.

Calculators

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. The paper is not intended to require 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.

Paper 2

Students must have access to a GDC at all times. However, not all questions will necessarily require the

use of the GDC. Regulations covering the types of GDC allowed are provided in Diploma Programme

Assessment procedures.

Paper 3

Students must have access to a GDC at all times. However, not all question parts will necessarily

require the use of the GDC. Regulations covering the types of GDC allowed are provided in Diploma

Programme Assessment procedures.

Formula booklet

Each student must have access to a clean copy of the formula booklet during the examination. It is the

responsibility of the school to download a copy from IBIS or the Programme Resource Centre and to

ensure that there are sufficient copies available for all students.

Awarding of marks

Marks are awarded for method, accuracy, answers and reasoning, including interpretation.

In papers 1, 2 and 3, full marks are not necessarily awarded for a correct answer with no working.

Answers must be supported by working and/or explanations (in the form of, for example diagrams,

graphs or calculations). Where an answer is incorrect, some marks may be given for correct method,

provided that this is shown by written working. All students should therefore be advised to show their

working.

Paper 1

Duration: 2 hours

Weighting: 30%

• This paper consists of section A, short-response questions, and section B, extended-response

questions.

• Students are not permitted access to any calculator on this paper.

Syllabus coverage

• Knowledge of all topics is required for this paper. However, not all topics are necessarily assessed in

every examination session.

Mark allocation

• This paper is worth 110 marks, representing 30% of the nal mark.

• Questions of varying levels of difficulty and length are set. Therefore, individual questions may not

necessarily each be worth the same number of marks. The exact number of marks allocated to each

question is indicated at the start of the question.

Section A

• This section consists of compulsory short-response questions based on the whole syllabus. It is

worth approximately 55 marks.

• The intention of this section is to assess students across the breadth of the syllabus. However, it

should not be assumed that the separate topics are given equal emphasis.

Question type

• A small number of steps are needed to solve each question.

• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of

these.

Section B

This section consists of a small number of compulsory extended-response questions based on the

whole syllabus. It is worth approximately 55 marks.

Individual questions may require knowledge of more than one topic.

The intention of this section is to assess students across the breadth of the syllabus in depth. The

range of syllabus topics tested in this section may be narrower than that tested in section A.

Question type

• Questions require extended responses.

• Individual questions will develop a single theme.

• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of

these.

• Normally, each question reects an incline of difficulty, from relatively easy tasks at the start

of a question to relatively difficult tasks at the end of a question. The emphasis is on sustained

reasoning.

Paper 2

Duration: 2 hours

Weighting: 30%

• This paper consists of section A, short-response questions, and section B, extended-response

questions.

• A GDC is required for this paper, but not every question will necessarily require its use.

Syllabus coverage

• Knowledge of all topics is required for this paper. However, not all topics are necessarily assessed in

every examination session.

Mark allocation

• This paper is worth 110 marks, representing 30% of the nal mark.

• Questions of varying levels of difficulty and length are set. Therefore, individual questions may not

necessarily each be worth the same number of marks. The exact number of marks allocated to each

question is indicated at the start of the question.

Section A

• This section consists of compulsory short-response questions based on the whole syllabus. It is

worth approximately 55 marks.

• The intention of this section is to assess students across the breadth of the syllabus. However, it

should not be assumed that the separate topics are given equal emphasis.

Question type

• A small number of steps are needed to solve each question.

• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of

these.

Section B

• This section consists of a small number of compulsory extended-response questions based on the

whole syllabus. It is worth approximately 55 marks.

• Individual questions may require knowledge of more than one topic.

• The intention of this section is to assess students across the breadth of the syllabus in depth. The

range of syllabus topics tested in this section may be narrower than that tested in section A.

Question type

• Questions require extended responses.

• Individual questions will develop a single theme.

• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of

these.

• Normally, each question reects an incline of difficulty, from relatively easy tasks at the start

of a question to relatively difficult tasks at the end of a question. The emphasis is on sustained

reasoning.

Paper 3

Duration: 1 hour

Weighting: 20%

• This paper consists of two compulsory extended-response problem-solving questions.

• A GDC is required for this paper, but not every question part will necessarily require its use.

Syllabus coverage

• Where possible, the rst part of each question will be on syllabus content leading to the problem-

solving context. Therefore, knowledge of all syllabus topics is required for this paper.

Mark allocation

• This paper is worth 55 marks, representing 20% of the nal mark.

• Questions may be unequal in terms of length and level of difficulty. Therefore, each question may

not be worth the same number of marks. The exact number of marks allocated to each question is

indicated at the start of each question.

Question type

• Questions require extended responses involving sustained reasoning.

• Individual questions will develop from a single theme where the emphasis is on problem solving

leading to a generalization or the interpretation of a context.

• Questions may be presented in the form of words, symbols, diagrams or tables, or combinations of

these.

• Normally, each question reects an incline in difficulty, from relatively easy at the start of a question

to relatively difficult tasks at the end of the question. The emphasis is on problem solving.

## Internal Assessments

Purpose of internal assessment

Internal assessment is an integral part of the course and is compulsory for both SL and HL students. It enables students to demonstrate the application of their skills and knowledge and to pursue their personal interests without the time limitations and other constraints that are associated with written examinations. The internal assessment should, as far as possible, be woven into normal classroom teaching and not be a separate activity conducted after a course has been taught.

The internal assessment requirements at SL and at HL is an individual exploration. This is a piece of written work that involves investigating an area of mathematics. It is marked according to five assessment criteria.

Guidance and authenticity The exploration submitted for internal assessment must be the student’s own work. However, it is not the intention that students should decide upon a title or topic and be left to work on the internal assessment component without any further support from the teacher. The teacher should play an important role during both the planning stage and the period when the student is working on the exploration.

It is the responsibility of the teacher to ensure that students are familiar with:

• the requirements of the type of work to be internally assessed

• the IB academic honesty policy available on the programme resource centre

• the assessment criteria; students must understand that the work submitted for assessment must address these criteria effectively.

Collaboration and teamwork

Collaboration and teamwork are a key focus of the approaches to teaching in the DP. It is advisable that the teacher uses the available class time to manage student collaboration. While working on their exploration students should be encouraged to work collaboratively in the various phases of the process,

for example: • generating ideas

• selecting the topic for their exploration

• sharing research sources

• acquiring the necessary knowledge, skills and understanding

• seeking peer feedback on their writing. The Approaches to teaching and learning (ATL) website on the programme resource centre provides an excellent source for developing collaborative skills in students.

Time allocation Internal assessment is an integral part of the mathematics courses, contributing 20% to the final assessment in the SL and the HL courses. This weighting should be reflected in the time that is allocated to teaching the knowledge, skills and understanding required to undertake the work, as well as the total time allocated to carry out the work. It is recommended that a total of approximately 10-15 hours of teaching time should be allocated to the work.

This should include:

• time for the teacher to explain to students the requirements of the exploration

• class time for students to work on the exploration and ask questions

• time for consultation between the teacher and each student

• time to review and monitor progress, and to check authenticity.

Requirements and recommendations

Students can choose from a wide variety of activities: for example, modelling, investigations and applications of mathematics. To assist teachers and students in the choice of a topic, a list of stimuli is available in the teacher support material. However, students are not restricted to this list.

In developing their explorations, students should aim to make use of mathematics learned as part of the course. The mathematics used should be commensurate with the level of the course–that is, it should be similar to that suggested in the syllabus. It is not expected that students produce work that is outside the syllabus–however, this will not be penalized. Ethical guidelines should be adhered to throughout the planning and conducting of the exploration. Further details are given in the Ethical practice in the Diploma Programme poster on the programme resource centre.

Presentation

The following details should be stated on the cover page of the exploration:

• title of the exploration

• number of pages. The references are not assessed. However, if they are not included in the final report it may be flagged in terms of academic honesty.

The specific purposes of the exploration are to:

• develop students’ personal insight into the nature of mathematics and to develop their ability to ask their own questions about mathematics

• provide opportunities for students to complete a piece of mathematical work over an extended period of time

• enable students to experience the satisfaction of applying mathematical processes independently

• provide students with the opportunity to experience for themselves the beauty, power and usefulness of mathematics

• encourage students, where appropriate, to discover, use and appreciate the power of technology as a mathematical tool

• enable students to develop the qualities of patience and persistence, and to reflect on the significance of their work

• provide opportunities for students to show, with confidence, how they have developed mathematically.

## 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.

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 their mathematics course if they have not submitted an exploration

Criterion A: Presentation (4 marks)

The “presentation” criterion assesses the organization and coherence of the exploration.

A coherent exploration is logically developed, easy to follow and meets its aim. This refers to the overall structure or framework, including introduction, body, conclusion and how well the different parts link to each other.

A well-organized exploration includes an introduction, describes the aim of the exploration and has a conclusion. Relevant graphs, tables and diagrams should accompany the work in the appropriate place and not be attached as appendices to the document.

Appendices should be used to include information on large data sets, additional graphs, diagrams and tables.

A concise exploration does not show irrelevant or unnecessary repetitive calculations, graphs or descriptions. The use of technology is not required but encouraged where appropriate. However, the use of analytic approaches rather than technological ones does not necessarily mean lack of conciseness, and should not be penalized. This does not mean that repetitive calculations are condoned

Criterion B: Mathematical communication (4 marks)

The “mathematical communication” criterion assesses to what extent the student has:

• used appropriate mathematical language (notation, symbols, terminology).

Calculator and computer notation is acceptable only if it is software generated.

Otherwise it is expected that students use appropriate mathematical notation in their work

• defined key terms and variables, where required

• used multiple forms of mathematical representation, such as formulae, diagrams, tables, charts, graphs and models, where appropriate

• used a deductive method and set out proofs logically where appropriate.

Criterion C: Personal engagement (3 marks)

The “personal engagement” criterion assesses the extent to which the student engages with the topic by exploring the mathematics and making it their own. It is not a measure of effort.

Personal engagement may be recognized in different ways. These include thinking independently or creatively, presenting mathematical ideas in their own way, exploring the topic from different perspectives, making and testing predictions.

Further (but not exhaustive) examples of personal engagement at different levels are given in the teacher support material (TSM).

There must be evidence of personal engagement demonstrated in the student’s work. It is not sufficient that a teacher comments that a student was highly engaged. Textbook style explorations or reproduction of readily available mathematics without the candidate’s own perspective are unlikely to achieve the higher levels.

Significant: The student demonstrates authentic personal engagement in the exploration on a few occasions and it is evident that these drive the exploration forward and help the reader to better understand the writer’s intentions.

Outstanding: The student demonstrates authentic personal engagement in the exploration in numerous instances and they are of a high quality. It is evident that these drive the exploration forward in a creative way. It leaves the impression that the student has developed, through their approach, a complete understanding of the context of the exploration topic and the reader better understands the writer’s intentions.

Criterion D: Reflection (3 marks)

The “reflection” 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.

Some ways of showing critical reflection are: considering what next, discussing implications of results, discussing strengths and weaknesses of approaches, and considering different perspectives.

Substantial evidence means that the critical reflection is present throughout the exploration. If it appears at the end of the exploration it must be of high quality and demonstrate how it developed the exploration in order to achieve a level 3.

Criterion E: : Use of mathematics—HL (6 marks)

The “Use of mathematics” HL criterion assesses to what extent students use relevant mathematics in the exploration.

Students are expected to produce work that is commensurate with the level of the course, which means it should not be completely based on mathematics listed in the prior learning. The mathematics explored should either be part of the syllabus, at a similar level or slightly beyond. However, mathematics of a level slightly beyond the syllabus is not required to achieve the highest levels.

A key word in the descriptor is demonstrated. The command term demonstrate means to make clear by reasoning or evidence, illustrating with examples or practical application. Obtaining the correct answer is not sufficient to demonstrate understanding (even some understanding) in order to achieve level 2 or higher.

For knowledge and understanding to be thorough it must be demonstrated throughout. Lines of reasoning must be shown to justify steps in the mathematical development of the exploration.

Relevant refers to mathematics that supports the development of the exploration towards the completion of its aim. Overly complicated mathematics where simple mathematics would suffice is not relevant.

The mathematics can be regarded as correct even if there are occasional minor errors as long as they do not detract from the flow of the mathematics or lead to an unreasonable outcome. Precise mathematics is error-free and uses an appropriate level of accuracy at all times.

Sophistication: To be considered as sophisticated the mathematics used should be commensurate with the HL syllabus or, if contained in the SL syllabus, the mathematics has been used in a complex way that is beyond what could reasonably be expected of an SL student. Sophistication in mathematics may include understanding and using challenging mathematical concepts, looking at a problem from different perspectives and seeing underlying structures to link different areas of mathematics.

Rigour involves clarity of logic and language when making mathematical arguments and calculations. Mathematical claims relevant to the development of the exploration must be justified or proven. Students are encouraged to use technology to obtain results where appropriate, but understanding must be demonstrated in order for the student to achieve level 1 or higher, for example merely substituting values into a formula does not necessarily demonstrate understanding of the results.

The mathematics only needs to be what is required to support the development of the exploration. This could be a few small elements of mathematics or even a single topic (or sub-topic) from the syllabus. It is better to do a few things well than a lot of things not so well. If the mathematics used is relevant to the topic being explored, commensurate with the level of the course and understood by the student, then it can achieve a high level in this criterion

For Extended Essay literature:

https://www.ibo.org/contentassets/5895a05412144fe890312bad52b17044/extended-essay-brief-2016-en.pdf

## Sample Papers and Marks scheme

Sample Explorations