Mathematics: Applications and Interpretations II

5 Credits

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Grade 12: Unit Four Calculus

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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: Calculus14 | P a g e

Unit 4 Calculus

Grade Level: 12

Description: Calculus describes rates of change between two variables and the accumulation of limiting areas. Understanding these rates of change allows us to model, interpret and analyze real-world problems and situations. Calculus helps us understand the behavior of functions and allows us to interpret the features of their graphs.

The aim of the standard level content in the calculus topic is to introduce students to the key concepts and techniques of differential and integral calculus and their use to approach practical problems.

Throughout this topic students should be given the opportunity to use technology such as graphing packages and graphing calculators to develop and apply their knowledge of calculus.

Suggested concepts embedded in this topic: Change, patterns, relationships, approximation, space, generalization

New Jersey Core Curriculum Content Standards (NJCCCS): F-IF.A, F-IF.B, MP 1-8.

NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

1. Recall, select and use their knowledge of limits and derivatives interpreted as the limit of a gradient function or an instantaneous rate of change in a variety of familiar and

How can the concept of a limit be used to understand the behaviors of functions?

How can you analyze functions for intervals of continuity or points of discontinuity using

Introduction to the concept of a limit.Estimation of the value of a limit from a table or graph.Not required: Formal analytic methods of

Activity: Limits and Continuity Desmos Activity

Polygraph: Continuity Desmos Activity

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

Texas Instruments TI84 Plus Graphing Display Calculator

IB Question bank

www.geogebra.com

www.Khanacademy.org

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

unfamiliar contexts.F-IF.A, F-IF.B.6

limits?

How can the derivative of a function be found using the limit of a difference quotient?

TOK: What value does the knowledge of limits have? Is infinitesimal behavior applicable to real life? Is intuition a valid way of knowing in mathematics?

calculating limits. Derivative interpreted

as gradient function and as rate of change.Forms of notation: dydx , f ' (x),

dVdr or

dsdt

for the first derivative.Informal understanding of the gradient of a curve as a limit.

Links to other subjects: Marginal cost, marginal revenue, marginal profit, market structures (economics); kinematics, induced emf and simple harmonic motion (physics); interpreting the gradient of a curve (chemistry)

Use of technology: Spreadsheets, dynamic graphing software and GDC should be used to explore ideas of limits, numerically

Homework and Classwork assignments based on daily lessons.

Aim 8: The debate over whether Newton or Leibnitz discovered certain calculus concepts; how the Greeks’ distrust of zero meant that Archimedes’ work did not lead to calculus.

International-mindedness: Attempts by Indian mathematicians (500-1000 CE) to explain division by zero.

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

and graphically. Hypotheses can be formed and then tested using technology

2. Recall, select and use their knowledge of derivatives to identify when its graph is increasing, decreasing or has a maximum or minimum value.

MP1-8, F-IF.A, F-IF.B.6

How can derivatives be used to analyze and sketch the key features of a function?

How can technology be used to display a graph and its derivative?

How is the graph of a function related to the graph of its derivative?

TOK: Is it possible for an area of knowledge to describe the world without transforming it?

Increasing and decreasing functions.

Graphical interpretation of f ' ( x )>0 , f ' ( x )=0∧f ' (x )<0.

Identifying intervals on which functions are increasing (f ' ( x )>0) or decreasing (f ' ( x )<0).

Values of x where the gradient of a curve is zero.Solution of f′(x)=0.Local maximum and minimum points.

Students should be able to use technology to generate f ' (x) given f (x), and find the solutions of f ' (x)=0.Awareness that the local maximum/minimum will not necessarily be the greatest/least value of the

Activity: Increasing and Decreasing Functions with Derivatives (Desmos)

Card Sort: Derivative Match by Desmos

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

Homework and Classwork assignments based on daily lessons.

Other contexts: Profit, area, volume, cost.Links to other subjects: Displacement-time and velocity-time graphs and simple harmonic motion graphs (physics).

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

function in the given domain.

3. Determine the derivative of a polynomial function. MP1-8, F-IF.A, F-IF.B

TOK: The seemingly abstract concept of calculus allows us to create mathematical models that permit human feats such as getting a man on the Moon. What does this tell us about the links between mathematical models and reality?

Derivative of f (x)=a xn is f ' (x)=a xn−1, n∈ZThe derivative of functions of the form f ( x )=a xn+b xn−1 ±… where all exponents are integers.

Activity: Sketchy Derivatives (Desmos)

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

Homework and Classwork assignments based on daily lessons.

4. Recall, select and use their knowledge of differentiation to interpret the meaning of a derivative within a problem involving the slope of a tangent line, the equation of a tangent line or the equation of a normal line.F-IF.A, F-IF.B, MP 1-8.

How can derivatives and limits be used to model real world phenomenon?

How can a derivate be interpreted given a verbal context?

TOK: In what ways has technology impacted how knowledge is produced and shared in mathematics? Does technology simply allow us to arrange existing knowledge in new and

Tangents and normals at a given point, and their equations.

Use of both analytic approaches and technology.

Links to other subjects: Instantaneous velocity and optics, equipotential surfaces (physics); price elasticity (economics).

Activity: Sketch the tangent and normal lines to a polynomial function at a specific x-coordinates by hand and with technology. Examples should include local maximum or minimum values.

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

Homework and Classwork assignments based on daily lessons.

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

different ways, or should this arrangement itself be considered knowledge?

5. Recall, select and use their knowledge of definite and indefinite integral skills, results and models in both real and abstract contexts to approximate or find the area under the curve of a function defined on a finite or infinite interval. Use technology where appropriate.

F-IF.A, F-IF.B, MP 1-8.

What is the relationship between integration and differentiation of a function?

What is the relationship between an indefinite integral and the area under the curve of a function?

How can technology be used to find the integral of a function?

How can integrals be applied to problems involving area under a curve?

How is the expression

∫a

b

f ' ( x )dx represent area

under the curve of f ( x )

Introduction to integration as anti-differentiation of functions of the form f ( x )=a xn+b xn−1 ± where n∈Z ,n ≠−1.

Students should be aware of the link between anti-derivatives, definite integrals and area.

Anti-differentiation with a boundary condition to determine the constant term.

Example: If dydx

=3 x2+x and

y=10 when x=1, then y=x3+12 x2+8.5.

Definite integrals using technology.

Area of a region enclosed by a curve y=f (x ) and the x-

Activity: Riemann Sums; Indefinite and Definite Integrals (Geogebra)

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

Homework and Classwork assignments based on daily lessons.

Other contexts: Irregular areas that are not described by mathematical functions, for example lakes.

Links to other subjects: Kinematics (Physics).

Use of technology: Use dynamic graphing software to calculate the approximate area under a curve and interpret its meaning.

Calculus Geogebra Tasks

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

on the interval a ≤ x≤ b?

Given the derivative of a function and a boundary condition, how can you determine the original function?

How can rectangles and trapezoids be used to estimate the area between the curve of f (x) and the x-axis?

axis, where f (x)>0.

Students are expected to first write a correct expression before calculating the area,

for example ∫2

6

(3 x2+4)dx.

The use of dynamic geometry or graphing software is encouraged in the development of this concept.

Approximating areas using the trapezoidal rule.

Given a table of data or a function, make an estimate for the value of an area using the trapezoidal rule, with intervals of equal width.

Link to: upper and lower bounds (SL1.6) and areas under curves (SL5.5).

Enrichment: Exploring other numerical integration techniques such as Simpson’s rule.

6. Investigate optimization problems, both abstract and real-world, involving

How can a derivative be related to the local maximum or minimum of a function?

What real world contexts

Optimization problems in context.

Examples: Maximizing profit, minimizing cost, maximizing volume for a

Activity:

Summative and Formative Assessments (Quizzes &

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

maximizing or minimizing cost, area, volume or other unfamiliar contexts, making conjectures, drawing conclusions based on differentiation of a function and testing their validity.F-IF.A, F-IF.B, MP 1-8.

involving local extrema can be modeled and solved using a derivative?

TOK: How can the rise in tax for plastic containers, for example plastic bags, plastic bottles etc be justified using optimization?

given surface area.

In examinations, questions on kinematics will not be set.

Other contexts: Efficient use of material in packaging.

Links to other subjects: Kinematics (physics); allocative efficiency (economics).

Tests) for each topic.

Homework and Classwork assignments based on daily lessons.

Enrichment: Investigate other applications of calculus including area between two functions, higher order derivatives, volumes of a revolution or the use of basic differential equations to model logistic growth (modeling population over time).

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Unit 4 Vocabulary

LimitContinuityLimit notationDerivativeGradient functionDifference quotientRate of changeIncreasing on an intervalDecreasing on an intervalStationary pointRelative maximum pointRelative minimum pointAbsolute or global extremaFirst derivative testLimit notationHorizontal and vertical asymptotesAverage rate of changeInstantaneous rate of change

Prime notationLeibniz notation Optimization problemsSecant lineTangent lines (and equations of)Normal lines (and equations of)Vertical Normal LinesDifferentiationPower ruleConstant ruleSum or difference ruleDerivative notationOptimizationTrapezoid rule

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TOK Connections

What value does the knowledge of limits have? Is infinitesimal behaviour applicable to real life? Is intuition a valid way of knowing in mathematics?

The seemingly abstract concept of calculus allows us to create mathematical models that permit human feats such as getting a man on the Moon. What does this tell us about the links between mathematical models and reality?

In what ways has technology impacted how knowledge is produced and shared in mathematics? Does technology simply allow us to arrange existing knowledge in new and different ways, or should this arrangement itself be considered knowledge?

Is it possible for an area of knowledge to describe the world without transforming it?

How can the rise in tax for plastic containers, for example plastic bags, plastic bottles etc be justified using optimization?

Euler was able to make important advances in mathematical analysis before calculus had been put on a solid theoretical foundation by Cauchy and others. However, some work was not possible until after Cauchy’s work. What does this suggest about the nature of progress and development in mathematics? How might this be similar/different to the nature of progress and development in other areas of knowledge?

Music can be expressed using mathematics. Does this mean that music is mathematical/that mathematics is musical?

What is the role of convention in mathematics? Is this similar or different to the role of convention in other areas of knowledge?

In what ways do values affect our representations of the world, for example in statistics, maps, visual images or diagrams?

How have notable individuals such as Euler shaped the development of mathematics as an area of knowledge?

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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.

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.

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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 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

Attempts by Indian mathematicians (500-1000 CE) to explain division by zero.

The successful calculation of the volume of a pyramidal frustrum by ancient Egyptians (the Egyptian Moscow mathematical papyrus).

Accurate calculation of the volume of a cylinder by Chinese mathematician Liu Hui; use of infinitesimals by Greek geometers; Ibn Al Haytham,

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the first mathematician to calculate the integral of a function in order to find the volume of a paraboloid.

Does the inclusion of kinematics as core mathematics reflect a particular cultural heritage? Who decides what is mathematics?

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