Mathematics: Applications and Interpretations II

5 Credits

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Grade 12: Unit Two Probability

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IB Learner ProfileIB Programs aim to develop internationally minded people who are striving to become:

Inquirers Their natural curiosity is nurtured. They acquire the skills necessary to conduct constructive inquiry and research, and become independent active learners. They actively enjoy learning and this love of learning will be sustained throughout their lives.

Knowledgeable They explore concepts, ideas and issues, which have global relevance and importance. In so doing, they acquire, and are able to make use of, a significant body of knowledge across a range of disciplines.

Critical thinkers They exercise initiative in applying thinking skills critically and creatively to approach complex problems and make reasoned decisions.

Communicators They understand and express ideas and information confidently and creatively in more than one language and in a variety of modes of communication.

Risk-takers They approach unfamiliar situations with confidence and forethought, and have the independence of spirit to explore new roles, ideas and strategies. They are courageous and articulate in defending those things in which they believe.

Principled They have a sound grasp of the principles of moral reasoning. They have integrity, honesty, a sense of fairness and justice and respect for the dignity of the individual.

Caring They show empathy, compassion and respect towards the needs and feelings of others. They have a personal commitment to action and service to make a positive difference to the environment and to the lives of others.

Open-minded Through an understanding and appreciation of their own culture, they are open to the perspectives, values and traditions of other individuals and cultures and are accustomed to seeking and considering a range of points of view.

Well-balanced They understand the importance of physical and mental balance and personal wellbeing for themselves and others. They demonstrate perseverance and self-discipline.

Reflective They consider their own learning and personal development. They are able to analyze their strengths and weaknesses in a constructive manner.

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Course DescriptionThis course recognizes the increasing role that mathematics and technology play in a diverse range of fields in a data-rich world. As such, it emphasizes the meaning of mathematics in context by focusing on topics that are often used as applications or in mathematical modelling. To give this understanding a firm base, this course also includes topics that are traditionally part of a pre-university mathematics course such as calculus and statistics.

The course makes extensive use of technology to allow students to explore and construct mathematical models. Mathematics: applications and interpretation will develop mathematical thinking, often in the context of a practical problem and using technology to justify conjectures. Students should enjoy seeing mathematics used in real-world contexts and to solve real-world problems.

International Baccalaureate Organization. (2019). Mathematics: applications and interpretation guide (First assessment 2021) [Curriculum Guide]. Wales, United Kingdom: International Baccalaureate Organization (UK) Ltd.

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# Topic Suggested Timing

1 Introduction to Trigonometry 6 weeks

2 Probability 10 weeks

3 Statistics 10 weeks

4 Introduction to Calculus 10 weeks

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Educational TechnologyStandards

8.1.8.A.1, 8.1.8.B.1, 8.1.8.C.1, 8.1.8.D.1, 8.1.8.E.1, 8.1.8.F.1

Technology Operations and Concepts Create professional documents (e.g., newsletter, personalized learning plan, business letter or flyer) using advanced features of a

word processing program.

Creativity and Innovation Synthesize and publish information about a local or global issue or event on a collaborative, web-based service.

Communication and Collaboration Participate in an online learning community with learners from other countries to understand their perspectives on a global problem

or issue, and propose possible solutions.

Digital Citizenship Model appropriate online behaviors related to cyber safety, cyber bullying, cyber security, and cyber ethics.

Research and Information Literacy Gather and analyze findings using data collection technology to produce a possible solution for a content-related or real-world

problem.           

Critical Thinking, Problem Solving, Decision Making Use an electronic authoring tool in collaboration with learners from other countries to evaluate and summarize the perspectives of

other cultures about a current event or contemporary figure.   

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21st Century Life & Career SkillsStandards:

9.1.8.A.1, 9.1.8.A.2, 9.1.8.B.1, 9.1.8.C.1, 9.1.8.C.2, 9.1.8.C.3, 9.1.8.D.2, 9.1.8.D.3, 9.3.8.B.3

Learning and Innovation Skills: Creativity and Innovation

Use multiple points of view to create alternative solutions.

Critical Thinking and Problem Solving Develop strategies to reinforce positive attitudes and productive behaviors that impact critical thinking and problem-solving skills. Implement problem-solving strategies to solve a problem in school or the community.

Communication and Collaboration Skills Determine an individual’s responsibility for personal actions and contributions to group activities. Demonstrate the use of compromise, consensus, and community building strategies for carrying out different tasks, assignments,

and projects. Model leadership skills during classroom and extra-curricular activities.

Cross-Cultural Understanding and Interpersonal Communication Demonstrate the ability to understand inferences. Use effective communication skills in face-to-face and online interactions with peers and adults from home and from diverse

cultures.  

Career Exploration Evaluate personal abilities, interests, and motivations and discuss how they might influence job and career selection.       

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Career Ready PracticesCareer Ready Practices describe the career-ready skills that all educators in all content areas should seek to develop in their students. They are practices that have been linked to increase college, career, and life success. Career Ready Practices should be taught and reinforced in all career exploration and preparation programs with increasingly higher levels of complexity and expectation as a student advances through a program of study.

CRP1. Act as a responsible and contributing citizen and employee Career-ready individuals understand the obligations and responsibilities of being a member of a community, and they demonstrate this understanding every day through their interactions with others. They are conscientious of the impacts of their decisions on others and the environment around them. They think about the near-term and long-term consequences of their actions and seek to act in ways that contribute to the betterment of their teams, families, community and workplace. They are reliable and consistent in going beyond the minimum expectation and in participating in activities that serve the greater good.

CRP2. Apply appropriate academic and technical skills. Career-ready individuals readily access and use the knowledge and skills acquired through experience and education to be more productive. They make connections between abstract concepts with real-world applications, and they make correct insights about when it is appropriate to apply the use of an academic skill in a workplace situation.

CRP3. Attend to personal health and financial well-being. Career-ready individuals understand the relationship between personal health, workplace performance and personal well-being; they act on that understanding to regularly practice healthy diet, exercise and mental health activities. Career-ready individuals also take regular action to contribute to their personal financial well-being, understanding that personal financial security provides the peace of mind required to contribute more fully to their own career success.

CRP4. Communicate clearly and effectively and with reason. Career-ready individuals communicate thoughts, ideas, and action plans with clarity, whether using written, verbal, and/or visual methods. They communicate in the workplace with clarity and purpose to make maximum use of their own and others’ time. They are excellent writers; they master conventions, word choice, and organization, and use effective tone and presentation skills to articulate ideas. They are skilled at interacting with others; they are active listeners and speak clearly and with purpose. Career-ready individuals think about the audience for their

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Career Ready Practicescommunication and prepare accordingly to ensure the desired outcome. CRP5. Consider the environmental, social and economic impacts of decisions. Career-ready individuals understand the interrelated nature of their actions and regularly make decisions that positively impact and/or mitigate negative impact on other people, organization, and the environment. They are aware of and utilize new technologies, understandings, procedures, materials, and regulations affecting the nature of their work as it relates to the impact on the social condition, the environment and the profitability of the organization.

CRP6. Demonstrate creativity and innovation. Career-ready individuals regularly think of ideas that solve problems in new and different ways, and they contribute those ideas in a useful and productive manner to improve their organization. They can consider unconventional ideas and suggestions as solutions to issues, tasks or problems, and they discern which ideas and suggestions will add greatest value. They seek new methods, practices, and ideas from a variety of sources and seek to apply those ideas to their own workplace. They take action on their ideas and understand how to bring innovation to an organization.

CRP7. Employ valid and reliable research strategies. Career-ready individuals are discerning in accepting and using new information to make decisions, change practices or inform strategies. They use reliable research process to search for new information. They evaluate the validity of sources when considering the use and adoption of external information or practices in their workplace situation.

CRP8. Utilize critical thinking to make sense of problems and persevere in solving them. Career-ready individuals readily recognize problems in the workplace, understand the nature of the problem, and devise effective plans to solve the problem. They are aware of problems when they occur and take action quickly to address the problem; they thoughtfully investigate the root cause of the problem prior to introducing solutions. They carefully consider the options to solve the problem. Once a solution is agreed upon, they follow through to ensure the problem is solved, whether through their own actions or the actions of others.

CRP9. Model integrity, ethical leadership and effective management. Career-ready individuals consistently act in ways that align personal and community-held ideals and principles while employing strategies to positively influence others in the workplace. They have a clear understanding of integrity and act on this understanding in every decision. They use a variety of means to positively impact the directions and actions of a team or organization, and they apply insights into human behavior to change

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Career Ready Practicesothers’ action, attitudes and/or beliefs. They recognize the near-term and long-term effects that management’s actions and attitudes can have on productivity, morals and organizational culture.

CRP10. Plan education and career paths aligned to personal goals. Career-ready individuals take personal ownership of their own education and career goals, and they regularly act on a plan to attain these goals. They understand their own career interests, preferences, goals, and requirements. They have perspective regarding the pathways available to them and the time, effort, experience and other requirements to pursue each, including a path of entrepreneurship. They recognize the value of each step in the education and experiential process, and they recognize that nearly all career paths require ongoing education and experience. They seek counselors, mentors, and other experts to assist in the planning and execution of career and personal goals.

CRP11. Use technology to enhance productivity. Career-ready individuals find and maximize the productive value of existing and new technology to accomplish workplace tasks and solve workplace problems. They are flexible and adaptive in acquiring new technology. They are proficient with ubiquitous technology applications. They understand the inherent risks-personal and organizational-of technology applications, and they take actions to prevent or mitigate these risks.

CRP12. Work productively in teams while using cultural global competence. Career-ready individuals positively contribute to every team, whether formal or informal. They apply an awareness of cultural difference to avoid barriers to productive and positive interaction. They find ways to increase the engagement and contribution of all team members. They plan and facilitate effective team meetings.

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Differentiated Instruction Accommodate Based on Students Individual Needs: Strategies

Time/General

Extra time for assigned tasks

Adjust length of assignment

Timeline with due dates for reports and projects

Communication system between home and school

Provide lecture notes/outline

Processing

Extra Response time

Have students verbalize steps

Repeat, clarify or reword directions

Mini-breaks between tasks

Provide a warning for transitions

Reading partners

Comprehension

Precise step-by-step directions

Short manageable tasks

Brief and concrete directions

Provide immediate feedback

Small group instruction

Emphasize multi-sensory learning

Recall

Teacher-made checklist

Use visual graphic organizers

Reference resources to promote independence

Visual and verbal reminders

Graphic organizers

Assistive Technology

Computer/whiteboard

Tape recorder

Spell-checker

Audio-taped books

Tests/Quizzes/Grading

Extended time

Study guides

Shortened tests

Read directions aloud

Behavior/Attention

Consistent daily structured routine

Simple and clear classroom rules

Frequent feedback

Organization

Individual daily planner

Display a written agenda

Note-taking assistance

Color code materials

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EnrichmentAccommodate Based on Students individual Needs: Strategies

Adaption of Material and Requirements

Evaluate Vocabulary

Elevated Text Complexity

Additional Projects

Independent Student Options

Projects completed individual or with Partners

Self Selection of Research

Tiered/Multilevel Activities

Learning Centers

Individual Response Board

Independent Book Studies

Open-ended activities

Community/Subject expert mentorships

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AssessmentsSuggested Formative/Summative Classroom Assessments

Timelines, Maps, Charts, Graphic Organizers

Teacher-created Unit Assessments, Chapter Assessments, Quizzes

Accountable Talk, Debate, Oral Report, Role Playing, Think Pair, and Share

Projects, Portfolio, Presentations, Prezi, Gallery Walks

Homework

Concept Mapping

Primary and Secondary Source analysis

Photo, Video, Political Cartoon, Radio, Song Analysis

Create an Original Song, Film, or Poem

Glogster to make Electronic Posters

Internal and External IB Assessments

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Interdisciplinary Connections English Language Arts

Journal Writing Close reading of industry-related content Create a brochure for a specific industry Keep a running word wall of industry vocabulary

Social Studies Research the history of a given industry/profession Research prominent historical individuals in a given industry/profession Use historical references to solve problems

World Language Translate industry-content Create a translated index of industry vocabulary Generate a translated list of words and phrases related to workplace safety

Math Research industry salaries for a geographic area and juxtapose against local cost of living Go on a geometry scavenger hunt Track and track various data, such as industry’s impact on the GDP, career opportunities or among of individuals currently occupying careers

Fine & Performing Arts Create a poster recruiting young people to focus their studies on a specific career or industry Design a flag or logo to represent a given career field

Science Research the environmental impact of a given career or industry Research latest developments in industry technology Investigate applicable-careers in STEM fields

Course: Applications and Interpretations SL Topic: Probability and Probability Distributions14 | P a g e

Unit 2 Probability

Grade Level: 12

Description: Probability enables us to quantify the likelihood of events occurring and so evaluate risk. Both statistics and probability provide important representations which enable us to make predictions, valid comparisons and informed decisions. These fields have power and limitations and should be applied with care and critically questioned, in detail, to differentiate between the theoretical and the empirical/observed. Probability theory allows us to make informed choices, to evaluate risk and to make predictions about seemingly random events. The aim of the standard level content in the statistics and probability topic is to introduce students to important concepts, techniques and representations used in statistics and probability and their meaningful application in the real world. Students should be given the opportunity to approach this topic in a practical way, to understand why certain techniques are used and to interpret the results. The use of technology such as simulations, spreadsheets, statistics software and statistics apps can greatly enhance this topic.

It is expected that most of the calculations required will be carried out using technology, but explanations of calculations by hand may enhance understanding. The emphasis is on choosing the most appropriate technique, and understanding and interpreting the results obtained in context.

In examinations students should be familiar with how to use the statistics functionality of allowed technology. Data sets will be considered to be the population unless otherwise stated.

Suggested concepts embedded in this topic:Quantity, validity, approximation, modelling, relationships, patterns.

New Jersey Core Curriculum Content Standards (NJCCCS): NJSLS. S.CP.A, S.CP.B, S.MD.A, S.MD.B, S.IC.A, S.IC.B, S.ID.A

MP1-8

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

1. Recall, select and use their knowledge of probability concepts of trial, outcome, equally likely outcomes, sample space (U) and events in a variety of familiar and unfamiliar contexts. S.CP.A, S.CP.B, S.MD.A

TOK: To what extent are theoretical and experimental probabilities linked? What is the role of emotion in our perception of risk, for example in business, medicine and travel safety?

How can you test theoretical probability using sample data?

What are reasonable outcomes of a simulation?

How can you determine if a model is consistent with the results of a simulation or experiment?

What is the difference between experimental probability and theoretical probability and how are they related to one another?

When is it appropriate to use a randomized experiment as opposed to a

Concepts of trial, outcome, equally likely outcomes, relative frequency, sample space (U) and event.The probability of an event A is

P ( A )= n( A)n(U )

.

The complementary events A and A′ (not A).Sample spaces can be represented in many ways, for example as a table or a list.Experiments using coins, dice, cards and so on, can enhance understanding of the distinction between experimental (relative frequency) and theoretical probability.

Activity: Shooting arrows at a target task

Summative and Formative Assessments (Quizzes & Tests) for each topic.

Homework and Classwork assignments based on daily lessons.

Other contexts: Actuarial studies and the link between probability of life spans and insurance premiums, government planning based on likely projected figures, Monte Carlo methods.

Links to other subjects: Theoretical genetics and Punnett squares (biology); the position of a particle (physics).

Aim 8: The ethics of gambling.

International-mindedness: The St Petersburg paradox; Chebyshev and Pavlovsky

Texas Instruments TI84 Plus Graphing Display Calculator

IB Question bank

www.geogebra.com

www.Khanacademy.org

www.illustrativemathem atics.org

www.illuminations.nctm .org

www.youtube.com

http://ocw.mit.edu

www.NJCTL.org

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

sample survey?

Why do we use simulations to support inferences about data?

How can you determine if a report is showing you misleading data or conclusions?

Simulations may be used to enhance this topic.

Expected number of occurrences.Example: If there are 128 students in a class and the probability of being absent is 0.1, the expected number of absent students is 12.8.

(Russian).

Use of technology: Computer simulations may be useful to enhance this topic.

2. Recall, select and use their knowledge of combined events, mutually exclusive events, independent events and conditional probability, results and models in both abstract and real-world contexts to solve problems.

S.IC.A, S.IC.B

.

TOK: Can calculation of gambling probabilities be considered an ethical application of mathematics? Should mathematicians be held responsible for unethical applications of their work?

Use of Venn diagrams, tree diagrams, sample space diagrams and tables of outcomes to calculate probabilities.

Combined events: P ( A∪B )=P ( A )+P ( B )−P (A ∩B).

Mutually exclusive events: P( A ∩ B)=0.The non-exclusivity of “or”.

Activity: Monte Carlo Simulations and Conditional Probability

Textbook practice problems

Aim 8: The gambling issue: use of probability in casinos. Could or should mathematics help increase incomes in gambling?

Summative and Formative Assessments (Quizzes & Tests) for each topic.

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

Conditional probability:

P( A∨B)=P ( A ∩B )P ( B )

.An alternate form of this is: P( A ∩ B)=P(B)P( A∨B).

Problems can be solved with the aid of a Venn diagram, tree diagram, sample space diagram or table of outcomes without explicit use of formulae. Probabilities with and without replacement.

Independent events: P( A ∩ B)=P( A)P (B).

Homework and Classwork assignments based on daily lessons.

3. Concept of discrete random variables and their probability distributions.Expected value (mean), E(X) for discrete data.

TOK: What do we mean by a “fair” game? Is it fair that casinos should make a profit?

What are the key properties of a probability distribution?

Concept of discrete random variables and their probability distributions.

Expected value (mean), E(X) for discrete data.

Applications.

Probability distributions

Activity: Aim 8: Why has it been argued that theories based on the calculable probabilities found in casinos are pernicious when applied to everyday life (for example, economics)?

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

Applications.

S.ID.A and MP1-8.

will be given in the following ways:

x 1 2 3 4 5

P(X=x) 0.1 0.2 0.3 0.4 0.5

P ( X=x )=118 (4+x ) for x∈ {1,2,3 };

E( X)=0 indicates a fair game where X represents the gain of a player.

Summative and Formative Assessments (Quizzes & Tests) for each topic.

Homework and Classwork assignments based on daily lessons.

Other contexts: Games of chance.

4. Recall, select and use their knowledge of mean and variance of the binomial distribution in a variety of familiar and unfamiliar contexts.

S.MD.A, S.MD.B & MP1-8.

TOK: What criteria can we use to decide between different models?

How can you calculate the expected value of a binomial distribution?

How is Pascal’s Triangle related to the binomial distribution?

Binomial distribution. Situations where the binomial distribution is an appropriate model.

Mean and variance of the binomial distribution.

In examinations, binomial probabilities should be found using available technology.

Not required: Formal proof of mean and

Activity: Find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices and find the expected grade under various grading schemes. Find the probability of scoring less than 30% once in 4 trials.

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

When is it appropriate to apply a Binomial Distribution?

How can a Binomial Distribution be calculated without a GDC?

What other distributions are related to the Binomial Distribution?

variance.

Link to: expected number of occurrences (SL4.5).

Optional Enrichment: Link to Binomial Expansion.

Summative and Formative Assessments (Quizzes & Tests) for each topic.

Homework and Classwork assignments based on daily lessons.

Aim 8: Pascal’s triangle, attributing the origin of a mathematical discovery to the wrong mathematician.

International-mindedness: The so-called “Pascal’s triangle” was known to the Chinese mathematician Yang Hui much earlier than Pascal.

Enrichment: Hypothesis testing using the binomial distribution.

5. Recall, select and use their knowledge of the Normal Distribution in a

What are the properties and representations of a normal distribution?

How can a GDC be used to

The normal distribution and curve.

Properties of the normal distribution.

Activities: Overloading lifts task

M&M Activity-Data Collection Project:

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

variety of familiar and unfamiliar contexts.S.MD.A, S.MD.B & MP1-8.

look-up the probability of an event in a normal distribution?

Aim 8: Why might the misuse of the normal distribution lead to dangerous inferences and conclusions?

TOK: To what extent can we trust mathematical models such as the normal distribution? How can we know what to include, and what to exclude, in a model?

Diagrammatic representation.

Awareness of the natural occurrence of the normal distribution.

Students should be aware that approximately 68% of the data lies between μ ± σ, 95% lies between μ ±2σ and 99.7% of the data lies between μ±3 σ.

Normal probability calculations.

Probabilities and values of the variable must be found using technology.

Inverse normal calculations

For inverse normal calculations mean and standard deviation will be given.

Not Required: transformation to the standardized normal variable z.

Census At School: https://ww2.amstat.org/censusatschool/

Summative and Formative Assessments (Quizzes & Tests) for each topic.

Homework and Classwork assignments based on daily lessons.

Links to other subjects: Normally distributed real-life measurements and descriptive statistics (sciences, psychology, environmental systems and societies)

International-mindedness: De Moivre’s derivation of the normal distribution and Quetelet’s use of it to describe l’homme moyen.

Enrichment: Create a diagram to compare and

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

contrast different distributions including the Binomial, Geometric, Normal and Poison distributions. Provide context for each.

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Unit 2 Vocabulary

ProbabilityEventExperimentRandom experimentTheoretical probabilityExpected valueRelative frequencyVenn diagramSample space Sample space diagramsTree diagramExperimental (empirical) probabilityComplementIntersection of eventsUnion of eventsUniversal setThe Addition RuleMutually exclusive eventsThe Product RuleIndependent eventsBias

Product rule for independent eventsConditional probabilityDependent eventsProbability tree diagram‘with replacement’ and repeated events Probability distributionsRandom variableDiscrete vs continuous random variableExpected value (mean for discrete data)Discrete dataBinomial distribution normal distributionBinomial experimentMean and Variance of a binomial distributionArea beneath the normal distribution curveStandard normal distributionInverse normal distribution

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TOK Connections

Could mathematics make alternative, equally true, formulae? What does this tell us about mathematical truths? Does the use of statistics lead to an over-emphasis on attributes that can be easily measured over those that cannot?

To what extent are theoretical and experimental probabilities linked? What is the role of emotion in our perception of risk, for example in business, medicine and travel safety?

Can calculation of gambling probabilities be considered an ethical application of mathematics? Should mathematicians be held responsible for unethical applications of their work?

What do we mean by a “fair” game? Is it fair that casinos should make a profit?

What criteria can we use to decide between different models?

To what extent can we trust mathematical models such as the normal distribution? How can we know what to include, and what to exclude, in a model?

Why might the misuse of the normal distribution lead to dangerous inferences and conclusions?

Contribution to the Development of Students’ Approaches to Learning Skills

Techniques to enhance the thinking skills of the students. Ask students to formulate a reasoned argument to support their opinion or conclusion. Give students time to think through their answers before asking them for a response. Set students a task which requires higher-order thinking skills such as analysis or evaluation. Build on specific prior tasks. Help students to make their thinking more visible, for example: use a strategy such as a thinking routine.

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Problem solving is central to learning mathematics and involves the 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 course, students will be expected to demonstrate the following:

Knowledge and understanding: Recall, select and use their knowledge of mathematical facts, concepts and techniques in a variety of familiar and unfamiliar contexts.

Problem solving: Recall, select and use their knowledge of mathematical skills, results and models in both abstract and real-world contexts to solve problems.

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.

Technology: Use technology accurately, appropriately and efficiently both to explore new ideas and to solve problems.

Reasoning: Construct mathematical arguments through use of precise statements, logical deduction and inference and by the manipulation of mathematical expressions.

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.

Contributions to the Development of the Attribute(s) of the Learner Profile

Students exercise initiative in applying thinking skills critically and creatively to recognize and approach complex problems and make reasoned ethical decisions. Inquiry approaches: investigate unfamiliar situations, both abstract and real-world, involving organizing and analyzing

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information or measurements, drawing conclusions, testing validity, and considering their scope and limitation. Reflective approaches: thoughtfully consider the world and our own ideas and experiences, extend what we learn in the classroom to life. The aims of all DP mathematics courses are to enable students to:1. develop a curiosity and enjoyment of mathematics, and appreciate its elegance and power2. develop an understanding of the concepts, principles and nature of mathematics3. communicate mathematics clearly, concisely and confidently in a variety of contexts4. develop logical and creative thinking, and patience and persistence in problem solving to instil confidence in using mathematics5. employ and refine their powers of abstraction and generalization6. take action to apply and transfer skills to alternative situations, to other areas of knowledge and to future developments in their local and global communities7. appreciate how developments in technology and mathematics influence each other8. appreciate the moral, social and ethical questions arising from the work of mathematicians and the applications of mathematics9. appreciate the universality of mathematics and its multicultural, international and historical perspectives10. appreciate the contribution of mathematics to other disciplines, and as a particular “area of knowledge” in the TOK course11. develop the ability to reflect critically upon their own work and the work of others12. independently and collaboratively extend their understanding of mathematics.

Contributions to the Development of International Mindedness

The St Petersburg paradox; Chebyshev and Pavlovsky (Russian).

The so-called “Pascal’s triangle” was known to the Chinese mathematician Yang Hui much earlier than Pascal.

De Moivre’s derivation of the normal distribution and Quetelet’s use of it to describe l’homme moyen.

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