Mathematics: Applications and Interpretations SL I

5 Credits

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Grade 11: Unit Four Geometry

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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 give thoughtful consideration to 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 is designed for students who enjoy describing the real world and solving practical problems using mathematics, those who are interested in harnessing the power of technology alongside exploring mathematical models and enjoy the more practical side of mathematics. It is intended to meet the needs of students whose interest in mathematics is more practical than theoretical but seek more challenging content. Applications and Interpretations is a course with an emphasis on applications of mathematics. It is for students with varied backgrounds and abilities. It offers students opportunities to learn important concepts and techniques and to gain an understanding of a wide variety of mathematical topics. It prepares students to be able to solve problems in a variety of settings, to develop more sophisticated mathematical reasoning and to enhance their critical thinking.

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# Topic Suggested Timing

1 Number and Algebra 8 weeks

2 Functions 1 (Linear, Quadratic and Exponential) 10 weeks

3 Functions 2 (Cubic, Sinusoidal and Modeling) 10 weeks

4 Geometry 7 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

DBQ, Essays, Short Answer

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: Geometry15 | P a g e

Unit 4 Geometry

Grade Level: 11

Description: Geometry allows us to quantify the physical world, enhancing our spatial awareness in two and three dimensions. This branch provides us with the tools for analysis, measurement and transformation of quantities, movements and relationships. This unit is designed to emphasize the study of the properties and applications of common geometric figures in two and three dimensions. Geometry stresses the ability to reason logically and to think critically. The aims of this unit are to develop the ability to draw clear diagrams in two dimensions, and to apply appropriate geometric techniques to problem-solving in two and three dimensions.

New Jersey Core Curriculum Content Standards (NJCCCS): HSG.G.MG.1, HSG.GMD.A1, GMD.A.2, HSG.GMD.A.3, CG.CO.9, G.GPE.5, G.MG.1, HSG.GMD.A.3, HSG.CO.A.1, HSG.C.A.2, HSG.C.A.3, HSG.C.B.5, G.C.5, F.TF..1, and MP 1-8.

NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

1. Calculate the distance between two points in three-dimensional space, and their midpoint, and calculate the size of an angle between two intersecting lines between a line and a plane.

HSG.G.MG.1,

How can diagrams be used to interpret solutions to real world problems?

How do changes in dimensions of a geometric figure affect area, surface area, and volume?

How can surface area and volume be used to find answers to real-world problems?

Distance formula in two-dimensions

Distance formula in three-dimensions

Midpoint formula in two-dimensions

Midpoint formula in three-dimensions

Calculate gradient

Identify positive, negative

Activity: Constructions

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

Homework and Classwork assignments based on daily lessons.

Texas Instruments TI84 Plus Graphing Display Calculator

IB Question bank

Geometer’s Sketchpad

Desmos Online Graphing Calculator

www.geogebra.com

www.Khanacademy.org www.illustrativemathematics.org

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

HSG.GMD.A.1, HSG.GMD.A.2,HSG.GMD.A.3 andMP 1-8

Why is it important to be able to relate two-dimensional drawings with three-dimensional figures

and zero gradientwww.illuminations..nctm.org

www.youtube.com

http://ocw.mit.edu

www.NJCTL.org

2. Utilize the properties of straight lines to determine the standard forms of linear equations; calculate the gradient and recognize the properties of parallel and perpendicular lines.

CG.CO.9, G.GPE.5MP 1-8

What are the building blocks of Geometry?How are geometry and algebra related to each other?

Why is there a relationship between the gradient of parallel and perpendicular lines?

How are angles and parallel and perpendicular lines used in real-world problems?

Distance formula

Midpoint formula

The gradient of a line.

Finding x and y intercepts.

Finding the equation of a straight line.

The gradient-intercept form.

Identify parallel and perpendicular lines.

Activity: Constructions

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

Homework and Classwork assignments based on daily lessons.

3. Understand the geometry of three-dimensional solids: right pyramid; right

What are various ways to describe the measures and properties of geometric shapes that

Recognize three dimensional solids.

Calculate the Volume and surface area of the three-

Activity: Dorm Dilemma

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

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

cone; sphere; hemisphere; and combinations of these solids.

HSG.G.MG.1, HSG.GMD.A.1, HSG.GMD.A.2,HSG.GMD.A.3 andMP 1-8

best represent a real-world object?

What are the attributes of each three-dimensional figure?

Why is it important to be able to relate two-dimensional drawings with three dimensional figures?

dimensional solids.

Calculate the distance between two points within a solid: between vertices, vertex to midpoint or from midpoint to midpoint.

Calculate the size of an angle between two lines or between a line and a plane.

Homework and Classwork assignments based on daily lessons.

4. Use a Voronoi diagram to partition a plane for a set of points into convex polygons such that each polygon contains exactly one generating point every point in a given polygon is closer to its generating point than to any other.MP 1-8

What does a Voronoi diagram show?

Why is a vertex always equidistant from three sites?

How many edges meet at a vertex of a Voronoi diagram?

Use perpendicular bisectors to fore the cells of a Voronoi diagram.

The cell boundaries of Voronoi diagrams are called a vertex, which is equidistant from the three surrounding sites.

Finding the nearest neighbor interpolation

Activity: Toxic Waste Dump

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

Homework and Classwork assignments based on daily lessons.

5. Use knowledge of circles to find the

How can you use circles to solve real world

Calculate the length of an arc in a sector or given an

Activity: Constructions

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NJDOE Student Learning Objective

Essential Questions Content Activities & Assessments

Resources

length of arc and the area of a sector.HSG.CO.A.1, HSG.C.A.2, HSG.C.A.3, G.C.5.F.TF.A.1 and MP 1-8

problems?

How do you find the length of an arc?

How do you find the area of a sector using arc measure and radians

area.

Calculate area of a sector

The circular radian measure of angles; length of an arc; area of a sector.

Find the measure of the angle given the length of the arc.

Assessments (Quizzes & Tests) for each topic.

Homework and Classwork assignments based on daily lessons.

6. Derive Equations of perpendicular bisectors.

HSG.C.B.5 and MP 1-8

What conjectures can you make about a point on the perpendicular bisector of a segment and a point on the bisector of an angle?

How can you use perpendicular bisectors to find the point that is equidistant from all the vertices of a triangle?

Use perpendicular bisectors to find measures

Use angle bisectors to find measures and distance relationships

Write equations for perpendicular bisectors.

Activity: Constructions

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

Homework and Classwork assignments based on daily lessons.

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Unit 4 Vocabulary

Point-gradient formStandard formInterceptsGradientSolidsArc sectorBisectorPerpendicular bisector

Platonic solidsPolyhedronFacesEdgesVerticesVoronoi diagramCellsSitesBoundaries/edges

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TOK Connections

It is often said that Euler’s relation, ei𝛑 +1=0, is the most beautiful equation in all of mathematics. What is meant by the beauty and elegance in mathematics?Mathematicians admire elegance and simplicity above all else, and the goal in solving a problem is to find the method that does the job in the most efficient manner ~Keith Devlin. What is elegance and simplicity in mathematical proof?What are platonic solids and why ate they an important part of the language of mathematics?Why are symbolic representations of three-dimensional objects easier to deal with than visual representations? What does this tell us about our knowledge of mathematics in other dimensions.Around the world you will often encounter different words for the same item, such as gradient and slope, trapezium and trapezoid, or root and surd. Sometimes more than one type of symbol might have the same meaning. To what extent does the language we use shape the way we think?Are algebra and geometry two separate domains of knowledge?

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.Organize and analyze information or measurements, drawing conclusions, testing validity, and considering their scope and limitations.

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

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Contributions to the Development of International Mindedness

Area can be measure in square units, but also in “packets” like hectares or acres. Which countries use bigha, mou, feddan, rai and tsubo?The stones used to make the pyramids in Egypt were constructed with a 13-knot rope replicating the shape of a 3-4-5 triangle to ensure sides meeting at right angles. Which do you think is superior: the Bourbaki group analytical approach other Mandelbrot visual approach to mathematics?

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