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Course Overview

Foundations of Mathematics, Grade 10, Applied

Identifying Information

Course Title: Foundations of Mathematics

Grade : 10

Course Type: Applied

Ministry Course Code: MFM2P

Credit Value: 1.0

Description/Rationale

This course enables students to consolidate their understanding of key mathematical concepts through hands-on activities and to extend their problem-solving experiences in a variety of applications. Students will solve problems involving proportional reasoning and the trigonometry of right triangles; investigate applications of piecewise linear functions; solve and apply systems of linear equations; and solve problems involving quadratic functions. The effective use of technology in learning and in solving problems will be a focus of the course.

How This Course Supports the Ontario Catholic School Graduate Expectations

Students will apply Christian values to pose and solve problems, to make logical decisions, and to become critical thinkers who share their abilities for the benefit of all in their classroom and school community. A supportive mathematics classroom provides a caring and sensitive environment where the dignity and value of all students is respected and affirmed as they grow in confidence in their mathematical abilities. Mathematical investigations will promote a respect for God’s creation and an understanding of the need to use resources wisely.

Unit Titles (Time + Sequence)

Unit 1	Modelling in Business and Finance	40 hours
Unit 2	Modelling with Quadratics	35 hours
Unit 3	Proportional Reasoning in Similarity and Applied Trigonometry	25 hours
Unit 4	Summative Assessment Activities	10 hours

Unit Organization

Unit 1: Modelling in Business and Finance

Time: 40 hours

Description

In this unit, students will use the world of finance as a basis for further exploration of linear models. Students will consolidate knowledge gained in the Grade 9 Applied or Academic course, and will examine situations that will lead them to the use of:

· piece-wise linear functions;

· graphical and algebraic solutions of linear systems;

· scale diagrams (e.g., distortion, technological drawings, costing, etc.);

· calculations involving rate, ratio, and percent;

· numerical investigation of optimization problems using technology to reveal a need for quadratics.

Ontario Catholic Graduate Expectations: CGE 1d, 2b, 2c, 3c, 4a, 4f, 5a,5b, 5g, 7b.

Strand(s): Linear Functions and Proportional Reasoning

Overall Expectations: All expectations from linear functions strand, PRV.01, QFV.03.

Specific Expectations: All expectations from the linear functions strand and PR1.01, PR1.02, PR1.03, QF1.01, QF3.02.

Unit 2: Modelling with Quadratics

Time: 35 hours

Description

In this unit, students will move from the numerical analysis of optimization problems to a graphical and algebraic approach involving quadratic functions and their applications. Students will progressively:

· use graphing technology to investigate and identify the effects of simple transformations on the graph of y = x²;

· develop skills in expanding, simplifying, and factoring quadratic and other algebraic expressions;

· solve problems related to various realistic situations by interpreting the graphs with the aid of technology.

Ontario Catholic Graduate Expectations: CGE2b, 2c, 3c, 3f, 4b, 4f, 5a.

Strand(s): Quadratic Functions

Overall Expectations: All expectations from the Quadratic Functions strand

Specific Expectations: All expectations from the Quadratic Functions strand as listed in the activities

Unit 3: Proportional Reasoning in Similarity and Applied Trigonometry

Time: 25 hours

Description

In this unit students will expand their proportional reasoning skills by examining the properties of similar triangles. In the context of applications, students will:

· investigate the properties of similar triangles using dynamic geometry software;

· solve realistic problems using similar triangles and the Pythagorean Theorem;

· define trigonometric formulas using ratios of sides of right triangles;

· use trigonometric ratios to solve a variety of problems;

· explore and describe the use of trigonometry in various occupations.

Ontario Catholic Graduate Expectations: CGE2b, 2c, 2e, 3c, 3e, 4b, 4e, 4f, 4g, 5a, 5f, 7i.

Strand(s): Proportional Reasoning

Overall Expectations: All expectations from the Proportional Reasoning strand

Specific Expectations: All expectations from the Proportional Reasoning strand as listed in the activities

Unit 4: Summative Assessment Activities

Time: 10 hours

Description

This course will conclude with a summative assessment unit incorporating a series of activities and a formal examination. The cumulative activities will consist of performance tasks that address learning expectations throughout the course. The formal examination will consist of a pencil and paper evaluation of knowledge and skills. The summative assessment unit will account for 30% of the final grade, with 20% allotted to the performance tasks and 10% allotted to the formal examination.

Ontario Catholic Graduate Expectations: CGE2b, 2c, 3c, 3b, 3e, 4f, 5a, 5g, 7b.

Overall Expectations: All overall expectations from each strand

Specific Expectations: Expectations as identified in the assessment activities

Course Notes

As in the Grade 9 program, mathematical modelling continues to be a primary focus of the Grade 10 Applied course. Skills are taught as the need occurs during investigations of rich contextual problems that lead to exploration using piece-wise linear, quadratic, and trigonometric models.

It is important to note that there are significant differences between the Grade 10 Applied and Academic expectations. For example in the Quadratic Functions strand, both courses contain algebraic and technological approaches, but the emphasis differs. In the Academic course, the emphasis is algebraic; in the Applied course, the emphasis is technological. In addition, students in the applied program have a greater opportunity to make connections between course content and the real world, especially in the areas of business and technology.

Teaching/Learning Strategies

In order to fully address the expectations in this course teachers will assume a variety of roles (including guide, facilitator, consultant, and instructor) and will employ a variety of strategies including:

· a balance of whole-class, small group, and individual instruction through student-centred and teacher-directed activities;

· the use of rich contextual problems which engage students and provide them with opportunities to demonstrate achievement of the course expectations;

· prompting, supporting, and challenging individual students and the class as a whole;

· approaches that will accommodate multiple learning styles (for example: provide verbal and written instructions as well as hands-on activities);

· the use of technological tools and software (e.g.,) graphing software, dynamic geometry software, internet, spreadsheets, multimedia, and computer-assisted design to facilitate the exploration and understanding of mathematical concepts;

· encouraging students to practice and extend their skills and knowledge outside the classroom in the form of field trips, external research, and appropriate guest speakers;

· the use of accommodations, remediation and/or extension activities where necessary to meet the needs of exceptional students.

Students will:

· develop increasing responsibility for their own learning;

· participate as active learners;

· be able to work individually and co-operatively;

· increase their ability to use technological aids for exploration of concepts;

· be accountable for pre-requisite skills.

Assessment/Evaluation Techniques

An effective assessment program in mathematics will include a balance of diagnostic, formative, and summative assessment instruments including the following:

To assess Knowledge and Understanding:

· unit tests

· quizzes

· final exam

· reports

· performance tasks

To assess Thinking/Inquiry/Problem Solving/Application skills in unfamiliar settings:

· performance assessment

· observation

· teacher/student conferences

To assess Communication skills:

· journals

· portfolios

· performance assessment

· observation

· presentations

· student-teacher conferences

To assess Application in familiar settings:

· tests

· quizzes

· performance assessment

Assessment Tools

· observational checklists

· performance checklists

· rubrics

· the Achievement Chart

· numeric marking

· rating scales

· peer evaluation

· self-evaluation

Resources

Videotapes, CD–ROMs, DVD

Life by the Numbers. PBS, 1998.

Software (Ministry Licensed)

ClarisWorks (spreadsheet)

Microsoft Works (spreadsheet)

Corel WordPerfect Suite (spreadsheet)

The Geometer’s Sketchpad™ (dynamic geometry software)

Zap-a-Graph (graphing software)

Math Trek (skills and concept development)

Web Sites

Extensive lists of mathematics sites can be found at:
http://sln.fi.edu/tfi/hotlists/math.html
http://forum.swarthmore.edu

Cornell University
http://www.tc.cornell.edu/Edu/MathSciGateway

Internet Public Library
http://www.ipl.org

Library of Congress
http://lcweb.loc.gov/homepage

National Council of Teachers of Mathematics
http://www.nctm.org

TV Ontario
http://www.tvo.org/osapac

Texas Instrument
http://www.ti.com/calc/docs

Satellite Images of Communities
www.terraserver.microsoft.com

Career Information
www.coolmath.com/careers.htm
http://on.cx.bridges.com

Books, Periodicals

NCTM Standards

NCTM Addenda Series

Mathematics in the Middle School (NCTM publication)

The Mathematics Teacher (NCTM publication)

Activities for Active Learning and Teaching (NCTM publication)

OAME Gazette

Exploring Geometry with Geometer’s Sketchpad. Key Curriculum Press.

Exploring Trigonometry with Geometer’s Sketchpad. Key Curriculum Press.

Graphic Algebra. Key Curriculum Press.

Moving Straight Ahead: Linear Relationships. Dale Seymour Publishing.

Accommodations

Teachers will refer to the student’s Individual Education Plan (IEP) and will consider the learning characteristics of their individual student to make necessary accommodations. Teachers should work in consultation with Resource Teachers, ESL/ELD Teachers, and parents to accommodate students as they work through the activities in order to achieve the expectations described in the IEP.

Accommodations for Exceptional Students

· opportunities for enrichment

· procedures, steps, and instructions in both written and oral form

· short simple instructions to provide detail

· additional time allowance for learning and assignment completion

· more concrete experience through use of appropriate technologies:

a) concrete materials and manipulatives

b) dynamic geometry software

c) graphing calculators

d) computer-assisted learning

· assignments presented to appeal to a variety of learning styles (visual, auditory, and kinesthetic)

· alternate formats for assignments:

a) written reports

b) oral presentations

c) audio/video taped reports and presentations

d) demonstrations

· co-operative group work, peer tutoring, and the buddy system

· scribe or photocopy student/teacher notes

· models provided for graphs, and diagrams

· posters/charts of skills posted in the classroom

· visual organizers

· opportunities to redo all or part of a task

Accommodations for Test and Exam Writing

· time extension

· language assistance (read questions, rephrase)

· technology use (computers, graphing calculators, concrete materials)

· isolated work environment

· de-stressed work environment

· physical accommodations (scribe, oral, or taped)

· oral/taped tests

· demonstrations

Accommodations for ESL/ELD

· reading levels appropriate to student abilities

· visual, interactive, and technological methods to facilitate learning of mathematics

· pairings or groupings with English speakers, or peer tutors

· mathematical terminology written on blackboard when using it

· key words and phrases highlighted

· lists of terminology provided before activity begins

· glossary of mathematical terms

· simplified language on handouts

· simplified instructions

· extra time to read, write, and complete assignments

· first language/English dictionaries for assignments/assessment

· electronic resources for preparation of assignments

· peer conferencing to reinforce instructions/information

· exposure to vocabulary and math terminology

· encouragement to students as they struggle to develop their written expression

· use of first language to access essential information and to discuss concepts before translating to English

OSS Policy Applications

The following list of resources will support many of the Ontario Secondary School Policies as well as the Ontario Catholic Secondary School Graduate Expectations:

Faith Development :

· Catholicity Across The Curriculum (Ontario Catholic School Trustees’ Association)

· Blueprints (Catholic Curriculum Cooperative - Central Region)

· This Moment of Promise (Ontario Conference of Catholic Bishops)

· Educating the Soul

Anti-Discrimination Education:

· refer to local board policies (e.g., Anti Racism and Ethno-Cultural Equity policy documents)

Equity/Social Justice Issues:

· refer to local board policies (e.g., Anti-harassment policies)

· refer to local school code of behaviour

Career Goals/Co-operative Education:

· Ontario Youth Apprenticeship Program

· Youth Employment Skills program

Community Partnerships:

· refer to local board policies (e.g., Relations with Business - Corporate Donations, Sponsorships, and Agreements)

Teachers should refer to local board policy documents for local interpretations. Teachers will be familiar with Ontario Secondary Schools, Grades 9 to 12, Program and Diploma Requirements, 1999. The Mathematics course of study allows students the opportunities for success. Modifying Curriculum Expectations as well as Alternative Curriculum Expectations may be planned to assist individual students.

Opportunities for career exploration occur throughout the course. In some school communities there may be a possibility for students who are interested in researching a topic (e.g., careers that use trigonometry) to job shadow and report back to the class. In other cases, work experience will be related to Career Exploration Activities El (Choices Into Action, Guidance and Career Education Program Policy for Ontario Elementary and Secondary Schools, 1999.) This course is designed to be flexible to suit the needs of all learners, in all communities.

Course Evaluation

Assessment and evaluation of student achievement provide teachers with an opportunity to think critically about their methods of instruction and the overall effectiveness of their program. Teachers may evaluate their course through a variety of methods. This course profile suggests a wide variety of strategies that include peer, self, and teacher evaluation. Both formative and summative methods should be used to gather information for reporting purposes. Assessment measures should also consider the personal reflections of students revealed through journal writing. Teachers should network (locally and provincially) to compare the effectiveness of various instructional strategies and assessment procedures and make the program changes needed to improve the achievement of their students. Feedback from the community (local, school, and business), may provide input to assist in making course improvements.

Coded Expectations, Foundations of Mathematics, MFM2P

Proportional Reasoning

Overall Expectations

PRV.01P

– solve problems derived from a variety of applications, using proportional reasoning;

PRV.02P

– solve problems involving similar triangles;

PRV.03P

– solve problems involving right triangles, using trigonometry.

Specific Expectations

Using Proportional Reasoning to Solve Problems from Applications

PR1.01P

– solve problems involving percent, ratio, rate, and proportion (e.g., in topics such as interest calculation, currency conversion, similar triangles, trigonometry, direct and partial variation related to linear functions) by a variety of methods and models (e.g., diagrams, concrete materials, fractions, tables, patterns, graphs, equations);

PR1.02P

– draw and interpret scale diagrams related to applications (e.g., technical drawings);

PR1.03P

– distinguish between consistent and inconsistent representations of proportionality in a variety of contexts (e.g., explain the distortion of figures resulting from irregular scales; identify misleading features in graphs; identify misleading conclusions based on invalid proportional reasoning).

Solving Problems Involving Similar Triangles

PR2.01P

– determine some properties of similar triangles (e.g., the correspondence and equality of angles, the ratio of corresponding sides) through investigation, using dynamic geometry software;

PR2.02P

– solve problems involving similar triangles in realistic situations (e.g., problems involving shadows, reflections, surveying);

PR2.03P

– define the formulas for the sine, the cosine, and the tangent of angles, using the ratios of sides in right triangles.

Solving Problems Involving the Trigonometry of Right Triangles

PR3.01P

– calculate the length of a side of a right triangle, using the Pythagorean theorem;

PR3.02P

– determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios;

PR3.03P

– solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation);

PR3.04P

– determine the height of an inaccessible object in the environment around the school, using the trigonometry of right triangles;

PR3.05P

– describe applications of trigonometry in various occupations.

Linear Functions

Overall Expectations

LFV.01P

– apply the properties of piecewise linear functions as they occur in realistic situations;

LFV.02P

– solve and interpret systems of two linear equations as they occur in applications;

LFV.03P

– manipulate algebraic expressions as they relate to linear functions.

Specific Expectations

Applying Piecewise Linear Functions

LF1.01P

– explain the characteristics of situations involving piecewise linear functions (e.g., pay scale variations, gas consumption costs, water consumption costs, differentiated pricing, motion);

LF1.02P

– construct tables of values and sketch graphs to represent given descriptions of realistic situations involving piecewise linear functions, with and without the use of graphing calculators or graphing software;

LF1.03P

– answer questions about piecewise linear functions by interpolation and extrapolation, and by considering variations on given conditions.

Interpreting Systems of Linear Equations

LF2.01P

– determine the point of intersection of two linear relations arising from a realistic situation, using graphing calculators or graphing software;

LF2.02P

– interpret the point of intersection of two linear relations within the context of a realistic situation;

LF2.03P

– solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination;

LF2.04P

– solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs.

Manipulating Algebraic Expressions

LF3.01P

– write linear equations by generalizing from tables of values and by translating written descriptions;

LF3.02P

– rearrange equations from the form y = mx + b to the form Ax + By + C = 0, and vice versa;

LF3.03P

– solve first-degree equations in one variable, including those with fractional coefficients, using an algebraic method;

LF3.04P

– isolate a variable in formulas involving first-degree terms.

Quadratic Functions

Overall Expectations

QFV.01P

– manipulate algebraic expressions as they relate to quadratic functions;

QFV.02P

– determine, through investigation, the relationships between the graphs and the equations of quadratic functions;

QFV.03P

– solve problems by interpreting graphs of quadratic functions.

Specific Expectations

Manipulating Algebraic Expressions

QF1.01P

– multiply two binomials and square a binomial;

QF1.02P

– expand and simplify polynomial expressions involving the multiplying and squaring of binomials;

QF1.03P

– describe intervals on quadratic functions, using appropriate vocabulary (e.g., greater than, less than, between, from... to, less than 3 or greater than 7);

QF1.04P

– factor polynomials by determining a common factor;

QF1.05P

– factor trinomials of the form x² + bx + c;

QF1.06P

– factor the difference of squares;

QF1.07P

– solve quadratic equations by factoring.

Investigating the Connection Between the Graphs and the Equations of Quadratic Functions

QF2.01P

– construct tables of values, sketch graphs, and write equations of the form y = ax² + b to represent quadratic functions derived from descriptions of realistic situations (e.g., vary the side length of a cube and observe the effect on the surface area of the cube);

QF2.02P

– identify the effect of simple transformations (i.e., translations, reflections, vertical stretch factors) on the graph and the equation of y = x², using graphing calculators or graphing software;

QF2.03P

– explain the role of a, h, and k in the graph of y = a(x - h)² + k;

QF2.04P

– expand and simplify an equation of the form y = a(x - h)² + k to obtain the form y = ax² + bx + c.

Solving Problems Involving Quadratic Functions

QF3.01P

– obtain the graphs of quadratic functions whose equations are given in the form = a(x - h)² + k or the form y = ax² + bx + c, using graphing calculators or graphing software;

QF3.02P

– determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software;

QF3.03P

– solve problems involving a given quadratic function by interpreting its graph (e.g., given a formula representing the height of a ball over elapsed time, graph the function, using a graphing calculator or graphing software, and answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball touch the ground? Over what interval is the height of the ball greater than 3 m?).

Ontario Catholic School Graduate Expectations

The graduate is expected to be:

A Discerning Believer Formed in the Catholic Faith Community who

CGE1a -illustrates a basic understanding of the saving story of our Christian faith;

CGE1b -participates in the sacramental life of the church and demonstrates an understanding of the centrality of the Eucharist to our Catholic story;

CGE1c -actively reflects on God’s Word as communicated through the Hebrew and Christian scriptures;

CGE1d -develops attitudes and values founded on Catholic social teaching and acts to promote social responsibility, human solidarity and the common good;

CGE1e -speaks the language of life... “recognizing that life is an unearned gift and that a person entrusted with life does not own it but that one is called to protect and cherish it.” (Witnesses to Faith)

CGE1f -seeks intimacy with God and celebrates communion with God, others and creation through prayer and worship;

CGE1g -understands that one’s purpose or call in life comes from God and strives to discern and live out this call throughout life’s journey;

CGE1h -respects the faith traditions, world religions and the life-journeys of all people of good will;

CGE1i -integrates faith with life;

CGE1j -recognizes that “sin, human weakness, conflict and forgiveness are part of the human journey” and that the cross, the ultimate sign of forgiveness is at the heart of redemption. (Witnesses to Faith)

An Effective Communicator who

CGE2a -listens actively and critically to understand and learn in light of gospel values;

CGE2b -reads, understands and uses written materials effectively;

CGE2c -presents information and ideas clearly and honestly and with sensitivity to others;

CGE2d -writes and speaks fluently one or both of Canada’s official languages;

CGE2e -uses and integrates the Catholic faith tradition, in the critical analysis of the arts, media, technology and information systems to enhance the quality of life.

A Reflective and Creative Thinker who

CGE3a -recognizes there is more grace in our world than sin and that hope is essential in facing all challenges;

CGE3b -creates, adapts, evaluates new ideas in light of the common good;

CGE3c -thinks reflectively and creatively to evaluate situations and solve problems;

CGE3d -makes decisions in light of gospel values with an informed moral conscience;

CGE3e -adopts a holistic approach to life by integrating learning from various subject areas and experience;

CGE3f -examines, evaluates and applies knowledge of interdependent systems (physical, political, ethical, socio-economic and ecological) for the development of a just and compassionate society.

A Self-Directed, Responsible, Life Long Learner who

CGE4a -demonstrates a confident and positive sense of self and respect for the dignity and welfare of others;

CGE4b -demonstrates flexibility and adaptability;

CGE4c -takes initiative and demonstrates Christian leadership;

CGE4d -responds to, manages and constructively influences change in a discerning manner;

CGE4e -sets appropriate goals and priorities in school, work and personal life;

CGE4f -applies effective communication, decision-making, problem-solving, time and resource management skills;

CGE4g -examines and reflects on one’s personal values, abilities and aspirations influencing life’s choices and opportunities;

CGE4h -participates in leisure and fitness activities for a balanced and healthy lifestyle.

A Collaborative Contributor who

CGE5a -works effectively as an interdependent team member;

CGE5b -thinks critically about the meaning and purpose of work;

CGE5c -develops one’s God-given potential and makes a meaningful contribution to society;

CGE5d -finds meaning, dignity, fulfillment and vocation in work which contributes to the common good;

CGE5e -respects the rights, responsibilities and contributions of self and others;

CGE5f -exercises Christian leadership in the achievement of individual and group goals;

CGE5g -achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others;

CGE5h -applies skills for employability, self-employment and entrepreneurship relative to Christian vocation.

A Caring Family Member who

CGE6a -relates to family members in a loving, compassionate and respectful manner;

CGE6b -recognizes human intimacy and sexuality as God given gifts, to be used as the creator intended;

CGE6c -values and honours the important role of the family in society;

CGE6d -values and nurtures opportunities for family prayer;

CGE6e -ministers to the family, school, parish, and wider community through service.

A Responsible Citizen who

CGE7a -acts morally and legally as a person formed in Catholic traditions;

CGE7b -accepts accountability for one’s own actions;

CGE7c -seeks and grants forgiveness;

CGE7d -promotes the sacredness of life;

CGE7e -witnesses Catholic social teaching by promoting equality, democracy, and solidarity for a just, peaceful and compassionate society;

CGE7f -respects and affirms the diversity and interdependence of the world’s peoples and cultures;

CGE7g -respects and understands the history, cultural heritage and pluralism of today’s contemporary society;

CGE7h -exercises the rights and responsibilities of Canadian citizenship;

CGE7i -respects the environment and uses resources wisely;

CGE7j -contributes to the common good

Acknowledgments

Identifying Information

Description/Rationale

How This Course Supports the Ontario Catholic School Graduate Expectations

Unit Titles (Time + Sequence)

Unit Organization

Unit 1: Modelling in Business and Finance

Unit 2: Modelling with Quadratics

Unit 3: Proportional Reasoning in Similarity and Applied Trigonometry

Unit 4: Summative Assessment Activities

Course Notes

Teaching/Learning Strategies

Assessment/Evaluation Techniques

Assessment Tools

Resources

Videotapes, CD–ROMs, DVD

Software (Ministry Licensed)

Web Sites

Books, Periodicals

Accommodations

Accommodations for Exceptional Students

Accommodations for Test and Exam Writing

Accommodations for ESL/ELD

OSS Policy Applications

Course Evaluation

Proportional Reasoning

Specific Expectations

Linear Functions

Specific Expectations

Quadratic Functions

Specific Expectations