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May 2014 subject reports Group 5, Mathematics HL TZ1

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Time zone variants of examination papers

To protect the integrity of the examinations, increasing use is being made of time zone variants of examination papers. By using variants of the same examination paper candidates in one part of the world will not always be taking the same examination paper as candidates in other parts of the world. A rigorous process is applied to ensure that the papers are comparable in terms of difficulty and syllabus coverage, and measures are taken to guarantee that the same grading standards are applied to candidates’ scripts for the different versions of the examination papers. For the May 2014 examination session the IB has produced time zone variants of Mathematics HL papers.

Internal assessment

Component grade boundaries

Grade: 1 2 3 4 5 6 7 Mark range: 0 - 2 3 - 5 6 - 8 9 - 11 12 - 14 15 - 16 17 - 20

The range and suitability of the work submitted

The majority of explorations were generally commensurate with the Maths HL content but the quality was very mixed with very few explorations in the top range. Unfortunately many explorations lacked citations. This requirement needs to be made clearly known to all teachers; otherwise students will risk a malpractice decision. Some of the explorations were too long, sometimes because the scope of the exploration was not focused enough. On the other hand a few explorations were too short and included very little mathematical content. Some repeated topics were seen like “The Monty Hall Problem”, “Rubic Cube Mathematics” or “Mathematics behind the Pokemon game”. A number of explorations were based on common textbooks problems and demonstrated little or superficial understanding of the mathematical concepts being explored. A few of the students however demonstrated thorough understanding and managed to personalize their explorations. Modelling explorations based on Physics problems were also abundant. The most popular topic explored was the “Parabolic Trajectory” and the “Catenary equation”.

Candidate performance against each criterion

A – In general students performed well against this criterion. Some teachers seem to believe that subheadings indicating “Aim”, “Rationale” etc., are required in order to achieve top levels. Most explorations were complete and concise, however, some were far too long. Works that were based on typical text book problems and depended a lot on sources tended to be incoherent and were difficult to follow. Any paraphrased information needs to be cited at the point in the exploration where it is used. A footnote referring to the bibliography is not enough and may lead to a decision of malpractice. B – Students did well in general on this criterion. Graphs and tables were often provided but not commented on. Sometimes graphs lacked labelling, and tables had no headings. The teacher sometimes condoned the misuse of computer notation; this lead to a change in the achievement level awarded. Some explorations lacked the definition of key terms used. C – This is the criterion that was mostly misinterpreted by teachers with a quite a few students being awarded top levels because of their commitment or enthusiasm for the subject without any of this


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being evident in the student work. Students who presented explorations based on common textbook problems beyond the HL curriculum, were unable to score highly on this criterion because the mathematics was not understood fully to enable them to take ownership and extend the work beyond the theory presented. Some teachers understood the criterion descriptors well and this was transmitted to students effectively. D – Some teachers misunderstood this criterion’s descriptors and must have conveyed to students that reflection was a summative of the work done. As such some explorations were written as an old “IA Task” with just a narrative about the scope and limitations of the work done and no meaningful or critical reflection. Again students who wrote a “textbook” problem investigation found it difficult to reflect on the process and / or results and their significance. For higher achievement levels in this criterion students need to consider further explorations, implications of results, compare the strengths and weaknesses of the different mathematical approaches of their investigation and also look at the topic from different perspectives. E – There was a large variety of mathematical content in the exploration, ranging from very basic mathematics to extensions well beyond the HL syllabus. A number of explorations were full of formulae which seemed to be copied from mathematical journals or Wikipedia without appropriate sources. It was not always clear whether the teacher had checked the mathematical content; this made it more difficult to understand how the achievement levels were interpreted and awarded by the teacher. In some explorations the content seemed “forced” and overly sophisticated abstract concepts were added in an attempt to raise the quality of the exploration. Often this created a patchwork of mathematical formulae and equations that were not necessarily understood by the student. Although an exploration may take the form of a research paper, containing mathematics that is found in appropriate sources, the student needs to demonstrate a deep understanding of the mathematics being explored. Recommendations and guidance for the teaching of future candidates The exploration should be introduced early in the course and referred to frequently enough to allow students to reflect on an area of Mathematics that best suits their interest and allows them to develop an appropriate exploration.

Students should be provided with material to stimulate ideas for the exploration. These may include movies, short videos, photographs, experiments etc…

Students need to develop research and writing skills through reading and understanding different forms of mathematical writing as well as the possible assignment of mini tasks.

Teachers should discuss the suitability of the topic chosen by students before a first draft is handed in.

Students should use some of the time allocated to the Exploration to explain clearly the expectations when it comes to using borrowed ideas from sources. Teachers need to make it very clear to students that each and every quoted, paraphrased, borrowed or stolen reference must be cited at the point of reference, otherwise the student’s work will be referred to the Academic Honesty department that may decide on a possible malpractice (plagiarizism).

The teacher should ensure that the work being submitted is the student’s own work.

The teacher must show evidence of checking the mathematics with tick marks, annotations and comments written directly on the students’ work. This will help the moderator to confirm the achievement levels awarded by the teacher.


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The teacher must mark a first draft of the exploration. This should provide students with written feedback. This should also lead to a discussion to ensure that the student understands the mathematics used and demonstrates this in the work.

Students should be discouraged from using difficult Mathematics beyond the HL syllabus if this cannot lead to some creativity or personalized problem.

Students should be reminded that the exploration should be between 6 to 12 pages typed in an appropriate font size (e.g. Arial 12). Diagrams and /or tables which are not significant and do not enhance the development of the exploration should not be included.

Candidates need to understand the difference between describing results and critically reflecting on their results.

Using difficult mathematics that goes well beyond the HL syllabus often results in a lack of thorough understanding and this in turn makes it difficult for the student to demonstrate Personal Engagement or Reflection.

Students should be encouraged to create their own questions based on their own individual interest which may include current social, economic or environmental problems in the community.

Teachers are encouraged to use past explorations (TSM exemplars) and engage students in marking them early on in the process. This will clarify the importance of each criterion and the impact the choice of topic may have on the achievement levels that may be reached.

Further Comments A number of explorations showed very little work other than paraphrasing entries in Wikipedia. It is the school’s responsibility to check for plagiarism before student work is submitted for assessment. When students choose to present an exploration which is based on a scientific phenomenon, they should be aware that they are writing about mathematics and not reproducing a laboratory report. It is felt that the new format of the IA has provided students with a great opportunity to explore a topic in Mathematics that they enjoy as well as take up ownership of their mathematical work. Paper one


Grade: 1 2 3 4 5 6 7 Mark range: 0 - 16 17 - 32 33 - 43 44 - 57 58 - 70 71 - 84 85 - 120

The areas of the programme and examination which appeared difficult for the candidates The sums and products of roots. This is a topic that it is new in the syllabus this year and was unfamiliar to many students. Some of the vector question (q 12) was poorly done, particularly surprising was how few knew what was required to prove a quadrilateral was a square.

May 2014 subject reports

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MATHEMATICS SL TZ1

Overall grade boundaries Grade: 1 2 3 4 5 6 7 Mark range:

0 - 17 18 - 34 35 - 48 49 - 60 61 - 71 72 - 83 84 - 100

Time zone variants of examination papers

To protect the integrity of the examinations, increasing use is being made of time zone variants of examination papers. By using variants of the same examination paper candidates in one part of the world will not always be taking the same examination paper as candidates in other parts of the world. A rigorous process is applied to ensure that the papers are comparable in terms of difficulty and syllabus coverage, and measures are taken to guarantee that the same grading standards are applied to candidates’ scripts for the different versions of the examination papers. For the May 2014 examination session the IB has produced time zone variants of Mathematics SL papers.

Internal assessment


Grade: 1 2 3 4 5 6 7 Mark range: 0 - 2 3 - 5 6 - 8 9 - 11 12 - 14 15 - 17 18 - 20


A wide range of appropriate topics with mixed quality was submitted. Candidates who had chosen appropriate topics could attain the upper levels of each criterion. A few others, however, produced work that was not commensurate with the level of this course.

Examples of popular themes that were received were card games and gambling, demographics, spread of disease, athletics/sports, and video games. In addition there were many candidates who attempted explorations on ‘common investigations’ or ‘textbook’ problems, like the Golden Ratio, Fibonacci numbers, the Birthday paradox, Monty Hall Problem, Pascal’s Triangle. There were also numerous modelling activities of real life situations that were very similar in style with the old portfolio modelling tasks. Often in these cases, candidates produced work that was a summary of common facts and or general history of the topic. This would normally demonstrate a lack of personal engagement. Nevertheless, explorations which were based on textbook problems sometimes did also

May 2014 subject reports Group 5, Mathematics SL TZ1

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lead to a good exploration, when the students decided to extend the investigation beyond the original problems and/or add something of their own in the exploration. However, this was not in abundance. Most of the explorations based on these common textbook problems or examples revolved around superficial understanding of the concepts, repetition of methods found on the internet and did not lend themselves to anything new, and hence, could not reach the highest levels.

The use of technology to develop regression functions in an attempt to model data was very common. In some cases this was done effectively with suitable mathematical support. However there were cases where the regression model was simply created and applied via technology with very little understanding shown. It is recommended that in future, students will justify their choice of regression model and reflect critically on their choice.

There were a few instances where candidates simply submitted explorations entirely based on old portfolio tasks, which were specifically designed for the old assessment criteria. As a result, such explorations would not necessarily provide the candidates with the opportunity to achieve the highest levels.

The students generally adhered to the recommendation that the exploration be between 6 and 12 pages long. However there were many that were too long. These were often found to be self-penalizing.


Criterion A:

This criterion was addressed well by most of the students, with work being coherent and organized to different extents. In general, they made an attempt to provide a relevant introduction, a rationale, an explicit aim and some sort of conclusion. They also tried to explain relevant concepts and took conciseness into consideration. However, some teachers did miss the subtle differentiation between level 3 and 4, which is about the conciseness and completeness of student work. For instance, very lengthy tables of data may be relegated to an appendix, with a summary in the text where the information is used. Similarly, pages after pages of repetitive calculations would affect the conciseness and flow of the paper; one or two sample calculations would suffice and the rest could be summarized in a table.

Students should be more careful with the stated aim matching what they write and present in their work, and the conclusion they reach. If they find themselves unable to write a paper originally intended because of space constraints or any other reasons, then it would be wise for them to adjust the intended aim accordingly.

In addition, work that depended on a lot of secondary sources tends to have less coherence and is more difficult to follow.

It is crucial that quoted information be correctly cited at the point in the exploration where it appears, both for the flow of the piece and for academic honesty.

Criterion B:

This criterion was appropriately dealt with in most of the explorations. There were plenty of varieties of mathematical presentation displayed but it is essential that students are reminded that these are to be appropriate. Most students were able to use appropriate terminology and notations in their work, including the use of appropriate ICT tools. However, there were students who still used inappropriate computer or calculator notation (this is not an issue if generated by the software), did not define key terms and employed inappropriate presentations, like poorly labelled diagrams or badly scaled graphs. Graphs copied from the Internet were inserted but without any real purpose. Graphs need a purpose and not just included to "use multiple forms of mathematical representation". Mathematical


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formulas and theorems just taken from the Internet were often included but did not always really add to the students’ work.

Criterion C:

This criterion proved to be the most difficult for teachers to assess or interpret and appeared to be the least understood by both students and teachers. It seems that too many teachers stated that they ‘saw’ engagement but this was not supported by the work submitted. They simply assumed that interest in the topic chosen by the student meant high personal engagement. High marks should not be awarded to students who just stated how much they enjoyed the topic or who demonstrated enthusiasm in class, unless this is seen in the exploration itself. Simply stating, "...it interests me..." is not personal engagement.

Students who explored a common investigation/textbook problem without any personal input or extension would not usually achieve the higher achievement levels in this criterion. Nevertheless, a number of teachers did seem to understand this criterion and were able to transmit that information to their students effectively. Some explorations do lend themselves more readily to high levels for personal engagement - for example those where students do their own research and data collection. With the more descriptive or historical topics it is not particularly easy to score highly here.

It is important to note that this criterion cannot be used to penalize late submission of work.

Criterion D:

This criterion was clearly understood well by many teachers and students and there was a wide range of achievement here. Many students simply described the results in their explorations. They also sometimes reflected on why they found the exploration interesting or enjoyed learning about it. Less often did they reflect on the analytical process of exploring. Many reflections were superficial. There were also cases where students would undertake explorations similar to the old portfolio tasks using the same questioning technique to reflect on the process. This did not lend itself to meaningful reflection.

Many students were under the impression that reflection could only come through in the conclusion and hence missed out on the opportunity of demonstrating substantial evidence of critical reflection throughout the exploration.

Higher levels were generally awarded to those students who considered further exploration, discussed implications of results, compared strengths and weaknesses of mathematical approaches and contemplated different perspectives.

Criterion E:

There was a wide range of achievement in this criterion from the lowest to the highest. Most students employed relevant mathematics that was commensurate with the level of the course, since the majority of topics were chosen to allow students to demonstrate at least ‘some’ mathematics. Students could be seen using many areas of mathematics from sequences to differential equations. In general, most teachers were able to determine whether or not the work was commensurate with the level of the course.

Regression analysis was used extensively but not with thorough understanding demonstrated. There were cases where the regression model was simply created and applied via technology with very little justification of their choice of regression model.

Teachers occasionally awarded high marks for work with numerous calculations even if no clear understanding was shown. In addition just showing the correct answer is not the same as showing understanding; it must be demonstrated. Students who did an excellent job of explaining their reasoning throughout their papers would gain the higher marks. A considerable number of students chose topics where the mathematics involved was clearly beyond their understanding. This was often


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because it had been taken from other sources. Although students did reference these sources, it was clear that many did not understand what they were doing mathematically.

The mathematics used need only be what is required to support the development of the exploration. This could be a few small topics or even a single topic from the syllabus. It needs to be made clearer to the students that it is better to do a few things well rather than trying to do more mathematics badly. If the mathematics used is relevant to the topic being explored, commensurate with the course, and understood by the student, then it can achieve a high level.

Recommendations and guidance for the teaching of future candidates x Teachers need to become familiar with the assessment criteria. There are numerous

examples located in the Teacher Support Material (TSM) on the Online Curriculum Centre (OCC), and they should familiarize themselves with the goals of the exploration, and how achievement levels for the new criteria are awarded by referring to the annotated student work in the TSM.

x One of the major issues was the lack of annotation and/or comments specific to individual student work provided by the teachers. In many cases, where comments were provided, these tended to be paraphrased from the descriptors. Teachers are reminded that it states in the TSM that one of their responsibilities is to assess the work accurately, annotating it appropriately to indicate where achievement levels have been awarded. This includes marking the mathematics and identifying any errors. These comments greatly help to confirm the level awarded by the teacher. Without supporting comments, changes are more likely.

x Teachers should provide relevant background information about their courses where appropriate.

x Teachers need to follow the procedures in the guide which allows students to submit a first draft. This way, teachers can assess the suitability of the topic, check the general organization and coherence, orally test the students’ knowledge of the mathematics and most importantly, ensure that the work is that of the student and not just a regurgitation of Wolfram, Wikipedia and other math sites.

x If students are going to type their work, they need to use an appropriate equation editor to avoid errors in notation. Students should not insert screenshots of equations and formulas from Wikipedia or Wolfram. This habit is a good indicator that the work is not their own. They would also benefit greatly from explicit instruction in the use of those tools.

x Teachers should discourage students from choosing common or textbook topics without a clear plan on how to make them into a proper exploration (with enough relevant mathematics in it) since these tended to be research based with very little personalization or creativity demonstrated. Yet teachers should not predefine certain topics for their students or limit them to a certain area of mathematics.

x Teachers need to emphasize the importance of clear referencing and proper citation in student work. Many students provided a bibliography or work cited page at the end of their documents without identifying how these resources have been used in the body of their work. Students should be guided to cite all resources used in the body of their work including all data and images from secondary sources.

x Teachers must annotate and write comments on student work. Those schools where teachers did this were more likely to have scores confirmed, as would be expected; the reasoning behind the scores awarded was readily apparent.

x Teachers could issue practice explorations to hone/practice certain specific skills. x Students should be reminded of the guidelines that the exploration should be between 6 and

12 pages long. x Schools should be strongly discouraged from mandating a particular type of exploration.

Rather students should be free to explore an area of their choice.

Further comments x There were instances of potential plagiarism, with many students using sources such as


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Wolfram or Wikipedia to copy content, formulas or ideas. x The mathematical content was often lacking even though the exploration itself was well

written and scored well. x The general feeling after seeing the variety of explorations is that this new IA has provided the

students with a great opportunity to explore what they wished, and given them the opportunity to appreciate mathematics in their own way.

x The exemplar materials and the frequently asked questions in the TSM have been/will be updated after the first live session. Teachers should make sure that they read these documents carefully, along with the updated guidance on applying the criteria.

Paper one


Grade: 1 2 3 4 5 6 7 Mark range: 0 - 17 18 - 34 35 - 46 47 - 56 57 - 65 66 - 75 76 - 90

General comments

The areas of the programme and examination which appeared difficult for the candidates

x Integration, definite and indefinite x Probability, combined and conditional x Application of discriminant x Direction vectors of lines x Infinite geometric series in a generalized context

The areas of the programme and examination in which candidates appeared well prepared

x Quadratic functions x Arithmetic sequences and series x Logarithms x Derivatives of polynomials x Intersection of lines in vector form x Probability on a tree diagram

The strengths and weaknesses of the candidates in the treatment of individual questions

Question 1: the quadratic function and its graph

Candidates showed great familiarity with the quadratic function and the vertex. A common error was to give a negative value for h . In solving for a, many found the substitution and subsequent algebra to be straightforward. Occasionally a candidate would expand the quadratic before substituting the given values, which led to a more burdensome expression.

November 2014 subject reports Group 5, Mathematics HL

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Internal assessment


Grade: 1 2 3 4 5 6 7

Mark range: 0 - 2 3 - 5 6 - 8 9 - 11 12 - 14 15 - 16 17 - 20


There was a wide range of interesting topics and some schools are to be commended on the guidance given to students. Tasks included modelling real life situations and others provided a synopsis of mathematics that was found during the research process. Among the latter, common explorations included the Birthday Problem, the Buffon needle, game theory, solving the Rubik’s cube and probabilities in a poker game. Although all these were suitable some topics did not allow students to show personal engagement apart from understanding the topic. Some explorations were also far too long, and sometimes too complex for peers to understand and also for the author to demonstrate good understanding of the mathematics involved.

It was also noted that some explorations were written in very small font to make them look concise, whereas others were very short with large diagrams, written in larger font and double-spaced.


Criterion A

In general students performed well in this criterion; however some explorations were far too long and therefore not concise. In some cases students had a rationale that was made up and not genuine. Some students did not write a clear aim and this often led to an incoherent piece of work that was not complete since they could not write a conclusion to put it all together.

Criterion B

Language, notation and terminology were generally correctly used. Students also used technology effectively and included diagrams, graphs and tables in the appropriate places, along the work. A single notational error can be condoned but repeated use throughout the exploration should be penalized. Marks were lost when students did a modelling exploration and did not define key terms.


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Criterion C

More candidates are aware of the requirements for this criterion, and some will give contrived reasons for choosing the topic thinking that this might give an extra mark. In many cases this rationale was not supported in the rest of the exploration. A number of explorations were mere reproductions of papers found on the Internet or common advanced textbook problems. In this case students would receive some marks for mastering new techniques but it is difficult to justify top achievement levels. Exploration in a mathematical topic occurs when candidates use something that has already been explained or systematized as a tool to approaching a new question that they themselves have come up with. For example, candidates are allowed to recreate a Koch snowflake as long as they generate their own fractal. Some teachers are still under the impression that Personal Engagement is a measure of effort so that the choice of achievement levels becomes subjective.

Criterion D

A number of students did not offer any meaningful or critical reflection but wrote a conclusion summarizing results. Teachers often marked high on this criterion, which then led to achievement levels being marked down by the moderator. Students and teachers need to understand that for high achievement levels reflection needs to take place throughout the exploration, through candidates isolating themselves from the problem, to see it from another point of view and to analyse its limits, the connections it may have to other similar problems, other implications that might arise, aspects that could have been considered but were not, applications to other real world problems or contexts, discussing the techniques being used, the validity of the results obtained etc.… Once more, students who reproduced common textbook problems presented work that contained only superficial reflection.

Criterion E

Mathematical content was varied and while some explorations contained extensive mathematics others were very descriptive. Most explorations were also in the form of a mini research paper with mathematical content being reproduced, sometimes without demonstrating good understanding. In some cases students demonstrated an understanding of the technique being used but no significant understanding as to why the particular technique works. Some modelling explorations incorporated fitting data into a model, without any justification for choosing that model or development of the mathematical model used. Students need to be well guided when choosing their topics. It is sometimes easier to achieve good levels using simple HL mathematics content than going beyond the syllabus.

Recommendations and guidance for the teaching of future candidates

The exploration needs to be introduced early on and referred to frequently with references to appropriate topics as they arise during the learning experience. Teachers must dedicate the number of hours specified in the programme to guide the candidates throughout the process. It is important that teachers highlight the importance of proper referencing and citations.


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Teachers are encouraged to read the new publications on the OCC; i.e., Academic Honesty in the IB educational context and Effective citing and referencing.

It is recommended that teachers provide candidates with mathematical stimuli in various forms, including but not limited to texts, web pages, specialized bibliographies, movies, video clips, photographs, paintings, graphic design, games of chance, board games, experiments, magic tricks, etc… The investigations that were part of the former internal assessment component may also serve as a source of ideas for candidates to develop their explorations creatively and constructively.

Students need to understand the 5 criteria thoroughly before starting to develop their own exploration. They need to be guided to choose a topic wisely; one that incorporates Mathematics that allows them to demonstrate full understanding, and which is commensurate with the course. The topic should allow them to demonstrate personal engagement by possibly giving them the avenue to be original and creative. The topic should also be focused and the aim achievable within the set number of pages (6 to 12).

Teachers should refrain from making anecdotal comments about student commitment to the exploration process, as this has no bearing on criterion C. Personal engagement should be evident in the students own work. All student work should contain evidence of marking with errors being pointed out;; this makes it easier for the moderator to confirm the teacher’s marks.

Further comments

Although it is not stated that teachers need to supply information regarding material before the exploration process was started, it is highly recommended that this background information be given as it helps the moderator put the development of the exploration into perspective.

It might be useful for candidates to practice assessing explorations that are available in the TSM. This gives them an idea of what is expected of them in a project of this nature and it will ensure that candidates understand all 5 criteria. Class time might even be spent developing a group exploration with the help of the teacher as a means of understanding the methodology involved.

Having said all this the moderation of Internal Assessment with the wide variety of creative and interesting explorations continues to be an enjoyable task for moderators.

November 2014 subject reports

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MATHEMATICS SL

Overall grade boundaries

Standard level

Grade: 1 2 3 4 5 6 7

Mark range: 0 - 17 18 - 36 37 - 49 50 - 61 62 - 73 74 - 85 86 - 100

Internal assessment


Grade: 1 2 3 4 5 6 7

Mark range: 0 - 2 3 - 5 6 - 8 9 - 11 12 - 14 15 - 17 18 - 20


A wide range of appropriate and engaging topics with mixed quality was submitted, leading to a range of marks from 1 to 20. Explorations that were suitable tended to have more original aims that had clear personal relevance and foci. Many students still submit explorations with research questions similar to textbook problems, or they were not focused enough to be dealt with adequately within 10 to 12 pages. For instance, there were still many attempts on topics like the Golden Ratio, Monty Hall Problem, Pascal’s Triangle, Handshake Problem and Koch Snowflake. These candidates generally produced work that was a summary of common facts and/or a general history of the topic. There were also many candidates who produced work that read like a common textbook example or explanation. In the Koch Snowflake, for example, student work mirrored the old IB Task, showing no personalization or extension. In both cases candidates tended to not score well.

The use of technology to develop regression functions in an attempt to model data was very common. Some schools included samples that were all modelling tasks and generally following the old portfolio style. Although there is nothing wrong with these tasks, per se, it would be disappointing if students felt limited to these or were specifically directed to do these by their teacher. In some cases these tasks were done effectively with suitable mathematical support. However there were cases where the regression model was simply created and

November 2014 subject reports Group 5, Mathematics SL

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applied via technology with very little understanding shown. It is hoped that students will be able to justify their choice of regression model and be able to reflect critically on their choice.

The students generally adhered to the suggestion that the exploration be between 6 and 12 pages long. However there were many that were very long. These were often found to be self-penalising.


Criterion A

Most students scored well on this criterion. Most of the work was well organized and systematically presented. They presented some sort of introduction, attempted an aim, made a conscious effort to organize the work and provided a conclusion at the end. Students who did poorly in this criterion usually did not have a focused aim and thus could not present a coherent development for the work. Also some students provided particularly contrived rationales or simply stated that they found the topic ‘interesting’, which is insufficient for a rationale. There were quite a few candidates that provided page after page of repetitive calculations that hurt the conciseness and flow of the paper. Students should provide only one or two sample calculations in the body and all other similar calculations should be summarized in a table. Coherence was generally good but at times the work flowed poorly, with missing explanations and poor linkage between subsections.

Criterion B

Presentation was generally done well. Most students had made a conscious effort to present their work appropriately and with a variety of mathematical presentations. They used an equation editor or other mathematical software to enter proper mathematical expressions. The use of appropriate diagrams with clear labelling was often a problem. It may be that tables and graphs are more easily generated by computer while diagrams take more effort. Many graphs and diagrams were cut and pasted from Internet sources and often these were without any real purpose. Graphs need a purpose and not just included to "use multiple forms of mathematical representation". Mathematical formulas and theorems just taken from the Internet were often included but did not always really add to the students’ work.

Criterion C

Many students made an effort to make the work their own by doing their own research, collecting their own data and providing convincing personal rationales for choosing the topics. On the other hand, there were quite a few students who did not make the exploration their own and only did descriptive work. Students who used textbook problems and basically cut-and-pasted from resources in the public domain often did poorly in this criterion. Similarly, there were still a good number of teachers awarding high marks for candidates who simply stated how much they enjoyed the topic or for the enthusiasm they demonstrated even though there was no evidence in the work of good personal engagement. It is important to note that this criterion cannot be used to penalise late submission of work.


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Criterion D

Many students could produce some reflections and attempted to make these meaningful. They would at least consider the relevance of the mathematics they were using or investigating. Unfortunately, only a few were capable of producing critical and substantial reflections throughout their explorations. Nevertheless, this did not stop some teachers from awarding the top level for student work which simply summarized the results.

Criterion E

There was a wide variety of mathematics used in the explorations and a wide range of levels of understanding. The majority of the students were able to produce explorations that are commensurate with the mathematics SL syllabus and relevant to the tasks. However often they were not able to show that they understood the concepts well. For instance, the mathematics appeared to be regurgitated from textbooks or the internet and not really applied to the question in hand. Applying it to the student’s own work needs to be encouraged. Only a few challenged themselves by going beyond the mathematics SL syllabus. The success rates of these attempts varied.

Students and teachers should be aware that just showing the correct answer is not the same as showing understanding, it must be demonstrated.

Recommendations and guidance for the teaching of future candidates

Teachers should ensure that they are familiar with all the relevant information in the guide and the Teacher Support Material (TSM), especially the teacher responsibilities in the TSM. In particular, the following points should be noted.

x Teachers should go over some of the exemplars in the TSM and mark them with their students so that the students will have a better idea on the expectations of each criterion.

x Teachers should encourage originality of work, particularly the idea of “making the work their own”. They should stress the idea of applying the mathematics that students have discovered to their own work.

x Students should be guided to cite all resources used in the body of the work. These include images and data that are used. A bibliography is not sufficient because it does not inform the reader how and where these resources have been used in the exploration.

x Students should be guided to produce explorations that have clear and focused aims, with evidence supporting their personal engagement.

x Teachers need to follow the suggested procedures in the TSM which allows students to submit a first draft. This way, teachers can assess the suitability of the topic, check the general organization and coherence, orally test the students’ knowledge of the mathematics and most importantly, ensure that the work is that of the student and not just a regurgitation of Wolfram, Wikipedia and other sites.

x Schools should be strongly discouraged from mandating a particular type of exploration. Rather students should be free to explore an area, where this leads to a


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decent exploration, of their choice. x It is extremely helpful to the moderation process if teachers annotate and comment

on the student work as well as on the form 5/EXCS. Teachers must indicate that they have checked mathematical processes, and noted if they are correct or not.

x Where there is more than one teacher, it is essential that internal standardization between teachers takes place.

Standard level paper one


Grade: 1 2 3 4 5 6 7

Mark range: 0 - 17 18 - 35 36 - 46 47 - 56 57 - 66 67 - 76 77 - 90

The areas of the programme and examination which appeared difficult for the candidates

x Integration using substitution and/or inspection x Expected value of a fair game x Sketching functions, including important features of the graph x Trigonometric ratios of obtuse angles x Conditional and binomial probability x Applying properties of logarithms x Vectors and vector equation of a line

The areas of the programme and examination in which candidates appeared well prepared

x Applying formulas for terms and sums of an arithmetic sequence x Sum of a probability distribution x Integration and differentiation of polynomial functions x Solving quadratic equations x Simple probability and tree diagrams

The strengths and weaknesses of the candidates in the treatment of individual questions

Question 1: quadratics

Parts (a) and (b) of this question were answered quite well by nearly all candidates, with only a few factoring errors in part (b). In part (c), although most candidates were familiar with the general parabolic shape of the graph, many placed the vertex at the y-intercept (0, -6), and very few candidates considered the endpoints of the function with the given domain.

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