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Course Profiles Catholic District School Board Writing Partnership Course Profile Principles of Mathematics Grade 10 Academic for teachers by teachers This sample course of study was prepared for teachers to use in meeting local classroom needs, as appropriate. This is not a mandated approach to the teaching of the course. It may be used in its entirety, in part, or adapted. April 2000 Course Profiles are professional development materials designed to help teachers implement the new Grade 10 secondary school curriculum. These materials were created by writing partnerships of school boards and subject associations. The development of these resources was funded by the Ontario Ministry of Education. This document reflects the views of the developers and not necessarily those of the Ministry. Permission is given to reproduce these materials for any purpose except profit. Teachers are also encouraged to amend, revise, edit, cut, paste, and otherwise adapt this material for educational purposes. Any references in this document to particular commercial resources, learning materials, equipment, or technology reflect only the opinions of the writers of this sample Course Profile, and do not reflect any official endorsement by the Ministry of Education or by the Partnership of School Boards that supported the production of the document. © Queen’s Printer for Ontario, 2000 Acknowledgments Catholic District School Board Writing Team – Mathematics – Academic Lead Board London District Catholic School Board in Partnership with Windsor Essex Catholic District School Board Course Profile Writing Team Mary Howe, Lead Writer, London District Catholic School Board Anne Marie Novacich, London District Catholic School Board Mary Rose Vanheule, London District Catholic School Board Doug St. Laurent, London District Catholic School Board Sue Trew, Dufferin Peel Catholic District School Board Sue Dilaudo, Windsor Essex Catholic District School Board Steve Chevalier, Windsor Essex Catholic District School Board Reviewers Margaret Sinclair, Toronto Catholic District School Board Paul Costa, Toronto Catholic District School Board Mary Steele, Wellington Catholic School Board Project Manager Mike Mitchell, London District Catholic School Board Thanks to: Dufferin Peel Catholic District School Board Toronto Catholic District School Board Wellington Catholic District School Board Frank Dipietro, Windsor Essex Catholic District School Board Ontario Association for Mathematics Education (OAME) Ontario Mathematics Co-ordinators Association (OMCA) Page 2 Principles of Mathematics - Academic Course Overview Principles of Mathematics, Grade 10, Academic Identifying Information Course Title: Principles of Mathematics Grade: 10 Course Type: Academic Ministry Course Code: MPM2D Credit Value: 1.0 Description/Rationale This course enables students to broaden their understanding of relations, extend their skill in multi-step problem solving, and continue to develop their abilities in abstract reasoning. Students will pursue investigations of quadratic functions and their applications; solve and apply linear systems; investigate the trigonometry of right and acute triangles; and develop supporting algebraic skills. How This Course Supports the Ontario Catholic School Board 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 Title (Time + Sequence) Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Modelling Linear Systems Analytic Geometry Modelling Quadratic Relations Similarity and Applied Trigonometry Summative Assessment Activities 13 hours 24 hours 41 hours 24 hours 8 hours Unit Descriptions Unit 1: Modelling Linear Systems Time: 13 hours In this unit, linear systems will be analysed both graphically and algebraically, with and without the use of technology. Activities in this unit provide a context for finding and interpreting points of intersection and lead students to solve linear systems by the methods of substitution and elimination. In preparation for unit two, students will explore a maximization problem that introduces the concept of quadratic functions and involves expanding and simplifying polynomial expressions. Ontario Catholic School Graduate Expectations: CGE1d, 2b, 2c, 4a, 4f, 5a, 5b, 5g, 7b. Strand(s): Analytic Geometry, Quadratic Functions Overall Expectations: AGV.01, QFV.03, QFV.04. Specific Expectations: AG1.01, AG1.02, AG1.03, QF3.02, QF4.02. Page 3 Principles of Mathematics - Academic Unit 2: Analytic Geometry Time: 24 hours Students will use analytic geometry to solve problems involving the properties of line segments and to verify geometric properties of triangles and quadrilaterals. Specific investigations will use these line segment properties to develop formulas for the lengths and midpoints of line segments; determine the equation of a circles centred at (0, 0); solve multi-step problems involving properties of line segments; determine the characteristics of triangles and quadrilaterals having fixed co-ordinates; investigate and verify geometric properties of triangles and quadrilaterals having fixed co-ordinates; and develop communication and problem-solving skills. Ontario Catholic School Graduation Expectations: CGE2b, 2c, 3c, 4b, 4c, 4f, 5e, 5g, 5h, 7b. Strand(s): Analytic Geometry Overall Expectations: AGV.02D, AGV.03D. Specific Expectations: AG2.01D, AG2.02D, AG2.03D, AG2.04D, AG3.01D, AG3.02D, AG3.03D. Unit 3: Modelling Quadratic Functions Time: 41 hours This unit will introduce, explore, and apply the properties of quadratic functions. Students will collect, analyse, manipulate and display data from primary and secondary sources to model quadratic relationships. Students will use graphing technology and paper and pencil tasks to explore the characteristics, equations, and graphs of quadratic functions. Realistic applications will be used to develop the quadratic model and its properties. Algebraic techniques of simplifying, factoring, and solving quadratic equations will be developed throughout the unit. Ontario Catholic School Graduate Expectations: CGE 2b, 2c, 3c, 3e, 4b, 4f, 5a, 5e, 5g, 7i, 7j. Strand(s): Quadratic Functions, Analytic Geometry Overall Expectations: All those from the Quadratic Functions Strand, AGV.01 Specific Expectations: All those from the Quadratic Functions Strand, AG1.01 Unit 4: Similarity and Applied Trigonometry Time: 24 hours In this unit, students will investigate the properties of similar and congruent triangles and their use in modelling realistic situations. Students will develop and investigate the primary trigonometric ratios using technology. Right-angled triangles will be used to measure the heights of inaccessible objects around the school. Students will apply trigonometric ratios, the sine law, and the cosine law to solve realistic problems in acute-angled triangles. Ontario Catholic School Graduate Expectations: CGE2b, 2c, 2e, 3c, 3e, 4b, 4e, 4f, 4g, 5a, 5f, 7i. Strand(s): Trigonometry Overall Expectations: All those from the Trigonometry Strand Specific Expectations: All those from the Trigonometry Strand Unit 5: Summative Assessment Activities Time: 8 hours This unit is made up of a series of performance tasks in which students will need to use all the knowledge and understanding of content and procedures of this course. The activities are based on the central theme of an amusement park. Teachers should choose the activities that will address as many of the expectations of the course as possible and still fit within their own time scheme. Any extra time should be used for other forms of examination review. Students also write a formal paper and pencil final exam. Page 4 Principles of Mathematics - Academic Note: Some of these activities may be used as final assessment instruments, final assessment review activities, or diagnostic tools. Teachers should combine a mixture of these activities along with a formal written exam in order to provide a comprehensive evaluation package. Ontario Catholic Graduate Expectations: CGE2b, 2c, 3c, 3f, 4a, 4f, 5a, 5e, 5g, 7b, 7i, 7j. Overall Expectations: All overall expectations from each strand Specific Expectations: All specific expectations from each strand Course Notes This course will involve students in rich and realistic applications that contain multi-step problem solving in all three strands: Quadratic Functions, Analytic Geometry, and Trigonometry. Due to the nature of the problems, the algebra in this course involves substantial complexity. Technology also plays an important role in the development of the course, through the use of graphing calculators and dynamic geometry software, for concept development and applications. The Quadratic Functions strand extends the Relationship Strand from the Grade 9 Academic course. The Quadratic Functions grows out of concrete experiments, modelling and technology with a gradual transition to the abstract algebraic treatment. The Analytic Geometry Strand has been sub-divided into two units: Modelling Linear Systems and Analytic Geometry. Modelling Linear Systems extends the knowledge and consolidates the understanding of Linear Relationships introduced in the Grade 9 Academic course. Both units are heavy in algebraic treatment and abstract ideas. This strand is a stepping stone towards the future study of the Grade 12 Geometry and Discrete Mathematics. Trigonometry is introduced using similarity and right triangles and is extended to acute triangles using the sine law and cosine law. Discussion of obtuse triangles will not be addressed in this course. However, obtuse triangles could be used as an extension to add enrichment to this topic if time permits. Due to the different expectations found in the Grade 9 Applied and Academic courses, students entering MPM2D from MFM1P will require reinforcement of some skills studied in less complexity in Grade 9, while some other concepts, not addressed in Grade 9, will need to be learned in order to ensure success in the course. Be aware that some review and consolidation of the following skills will be necessary for some students: rearrange the equation of a line from the form y = mx + b to the form Ax + By + C = 0 and vice versa. y 2 y1 ; m = -A/B. x 2 x1 develop and use these formulas for slope: y/x; determine the equation of a line given an complex description of the line that requires an multi-step solution (e.g., a line parallel to a given line and having the same x-intercept as another given line) identify and state restrictions on the variables in a relation describe a situation that could be modelled by a given linear equation determine the point of intersection of two linear relations using graphing technology and interpret the intersection point in the context of an application apply the exponent rules in expressions involving two variables common factor an expression solve equations with rational coefficients 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; Page 5 Principles of Mathematics - Academic the use of rich contextual problems which engage students, promote Catholic values, and provide students 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, computer-assisted design) to facilitate the exploration and understanding of mathematical concepts; encouraging students to practise and extend their skill 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 Page 6 Principles of Mathematics - Academic Assessment Tools observational checklists performance checklists rubrics the Achievement Chart marking schemes rating scales peer assessment self-assessment Accommodations Teachers will refer to the student Individual Education Plan (IEP) and will consider the learning characteristics of their individual students 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, instruction 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 (concrete materials and manipulatives; dynamic geometry software; graphing calculators; and computer-assisted learning) assignments presented to appeal to a variety of learning styles (visual, auditory, kinesthetic) alternate formats for assignments (written reports, oral presentations, audio/visual taped reports, presentations, and demonstrations) co-operative group work, peer tutor, buddy system scribe or photocopy student/teacher notes models provided for graphs, diagrams; posters/charts of skills posted in 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 physical accommodations (scribe, oral, taped); oral/taped tests Accommodations for ESL/ELD reading levels appropriate to student abilities visual, interactive, and technological methods to facilitate learning of mathematics pairings or grouping with English speakers, peer conferencing to reinforce instructions/information mathematical terminology written on the board 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 Page 7 Principles of Mathematics - Academic 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 Resources Print Activities for Active Learning and Teaching. (NCTM publication) Exploring Geometry with Geometer’s Sketchpad. Key Curriculum Press. Exploring Trigonometry with Geometer’s Sketchpad. Key Curriculum Press. Graphic Algebra. Key Curriculum Press. Mathematics in the Middle School. (NCTM publication) The Mathematics Teacher. (NCTM publication) Moving Straight Ahead: Linear Relationships. Dale Seymour Publishing. NCTM Addenda Series NCTM Standards OAME Gazette Software Ministry Licensed for Ontario schools: ClarisWorks (spreadsheet) Corel WordPerfect Suite (spreadsheet) The Geometer’s Sketchpad (dynamic geometry software) Math Trek (skills and concept development) Microsoft Works (spreadsheet) Zap-a-Graph (graphing software) Web Sites Extensive lists of mathematics sites can be found at: http://sln.fi.edu/tfi/hotlists/math.html http://forum.swarthmore.edu Career Information www.coolmath.com/careers.htm http://on.cx.bridges.com 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 Satellite Images of Communities www.terraserver.microsoft.com Texas Instruments http://www.ti.com/calc/docs Page 8 Principles of Mathematics - Academic Video Life by the Numbers. PBS, 1998. 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: Ontario Catholic Secondary School Graduate Expectations (Institute for Catholic Education) Catholicity Across The Curriculum (Ontario Catholic School Trustee’s Association) Blueprints (Catholic Curriculum Co-operative – Central Region) 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. The focus of job shadowing and career awareness may impact on the Trigonometry Unit. 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 (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 also provide input to assist in making course improvements. Page 9 Principles of Mathematics - Academic Coded Expectations, Principles of Mathematics, MPM2D Quadratic Functions Overall Expectations QFV.01D – solve quadratic equations; QFV.02D – determine, through investigation, the relationships between the graphs and the equations of quadratic functions; QFV.03D – determine, through investigation, the basic properties of quadratic functions; QFV.04D – solve problems involving quadratic functions. Specific Expectations Solving Quadratic Equations QF1.01D – expand and simplify second-degree polynomial expressions; QF1.02D – factor polynomial expressions involving common factors, differences of squares, and trinomials; QF1.03D – solve quadratic equations by factoring and by using graphing calculators or graphing software; QF1.04D – solve quadratic equations, using the quadratic formula; QF1.05D – interpret real and non-real roots of quadratic equations geometrically as the x-intercepts of the graph of a quadratic function. Investigating the Connections Between the Graphs and the Equations of Quadratic Functions QF2.01D – 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; QF2.02D – explain the role of a, h, and k in the graph of y = a(x - h)2 + k; QF2.03D – express the equation of a quadratic function in the form y = a(x - h)2 + k, given it in the form y = ax2 + bx + c, using the algebraic method of completing the square in situations involving no fractions; QF2.04D – sketch, by hand, the graph of a quadratic function whose equation is given in the form y = ax2 + bx + c, using a suitable method [e.g., complete the square; locate the x-intercepts if the equation is factorable; express in the form y = ax(x - s) + t to locate two points and deduce the vertex]. Page 10 Principles of Mathematics - Academic Investigating the Basic Properties of Quadratic Functions QF3.01D – collect data that may be represented by quadratic functions, from secondary sources (e.g., the Internet, Statistics Canada), or from experiments, using appropriate equipment and technology (e.g., scientific probes, graphing calculators); QF3.02D – fit the equation of a quadratic function to a scatter plot, using an informal process (e.g., a process of trial and error on a graphing calculator), and compare the results with the equation of a curve of best fit produced by using graphing calculators or graphing software; QF3.03D – describe the nature of change in a quadratic function, using finite differences in tables of values, and compare the nature of change in a quadratic function with the nature of change in a linear function; QF3.04D – report the findings of an experiment in a clear and concise manner, using appropriate mathematical forms (e.g., written explanations, tables, graphs, formulas, calculations), and justify the conclusions reached. Solving Problems Involving Quadratic Functions QF4.01D – determine the zeros and the maximum or minimum value of a quadratic function, using algebraic techniques; QF4.02D – determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software; QF4.03D – solve problems related to an application, given the graph or the formula of a quadratic function (e.g., given a quadratic function representing the height of a ball over elapsed time, 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?). Analytic Geometry Overall Expectations AGV.01D – model and solve problems involving the intersection of two straight lines; AGV.02D – solve problems involving the analytic geometry concepts of line segments; AGV.03D – verify geometric properties of triangles and quadrilaterals, using analytic geometry. Specific Expectations Using Linear Systems to Solve Problems AG1.01D – determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation; Page 11 Principles of Mathematics - Academic AG1.02D – solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination; AG1.03D – 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. Solving Problems Involving the Properties of Line Segments AG2.01D – determine formulas for the midpoint and the length of a line segment and use these formulas to solve problems; AG2.02D – determine the equation for a circle having centre (0, 0) and radius r, by applying the formula for the length of a line segment; identify the radius of a circle of centre (0, 0), given its equation; and write the equation, given the radius; AG2.03D – solve multi-step problems, using the concepts of the slope, the length, and the midpoint of line segments (e.g., determine the equation of the right bisector of a line segment, the coordinates of whose end point are given; determine the distance from a given point to a line whose equation is given; show that the centre of a given circle lies on the right bisector of a given chord); AG2.04D – communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions. Using Analytic Geometry to Verify Geometric Properties AG3.01D – determine characteristics of a triangle whose vertex coordinates are given (e.g., the perimeter; the classification by side length; the equations of medians, altitudes, and right bisectors; the location of the circumcentre and the centroid); AG3.02D – determine characteristics of a quadrilateral whose vertex coordinates are given (e.g., the perimeter; the classification by side length; the properties of the diagonals; the classification of a quadrilateral as a square, a rectangle, or a parallelogram); AG3.03D – verify geometric properties of a triangle or quadrilateral whose vertex coordinates are given (e.g., the line joining the midpoints of two sides of a triangle is parallel to the third side; the diagonals of a rectangle bisect each other). Page 12 Principles of Mathematics - Academic Trigonometry Overall Expectations TRV.01D – develop the primary trigonometric ratios, using the properties of similar triangles; TRV.02D – solve trigonometric problems involving right triangles; TRV.03D – solve trigonometric problems involving acute triangles. Specific Expectations Developing the Primary Trigonometric Ratios TR1.01D – determine the properties of similar triangles (e.g., the correspondence and equality of angles, the ratio of corresponding sides, the ratio of areas) through investigation, using dynamic geometry software; TR1.02D – describe and compare the concepts of similarity and congruence; TR1.03D – solve problems involving similar triangles in realistic situations (e.g., problems involving shadows, reflections, surveying); TR1.04D – 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 TR2.01D – determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios; TR2.02D – solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation); TR2.03D – determine the height of an inaccessible object in the environment around the school, using the trigonometry of right triangles. Solving Problems Involving the Trigonometry of Acute Triangles TR3.01D – determine, through investigation, the relationships between the angles and sides in acute triangles (e.g., the largest angle is opposite the longest side; the ratio of side lengths is equal to the ratio of the sines of the opposite angles), using dynamic geometry software; TR3.02D – calculate the measures of sides and angles in acute triangles, using the sine law and cosine law; TR3.03D – describe the conditions under which the sine law or the cosine law should be used in a problem; TR3.04D – solve problems involving the measures of sides and angles in acute triangles; TR3.05D – describe the application of trigonometry in science or industry. Page 13 Principles of Mathematics - Academic 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. Page 14 Principles of Mathematics - Academic 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; Page 15 Principles of Mathematics - Academic 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. Page 16 Principles of Mathematics - Academic Unit 1: Modelling Linear Systems Time: 13 hours Unit Description In this unit, linear systems will be analysed both graphically and algebraically, with and without the use of technology. Activities in this unit provide a context for finding and interpreting points of intersection and lead students to solve linear systems by the methods of substitution and elimination. In preparation for unit two, students will explore a maximization problem that introduces the concept of quadratic functions and involves expanding and simplifying polynomial expressions. Strand(s) and Expectations Ontario Catholic School Graduate Expectations: CGE1d, 2b, 2c, 4a, 4f, 5a, 5b 5.g, 7b. Strand(s): Analytic Geometry, Quadratic Functions Overall Expectations: AGV.01D, QFV.04. Specific Expectations: AG1.01D, AG1.02D, AG1.03D, QF1.01D, QF4.02D. Activity Titles (Time + Sequence) The activities in this unit, which is an extension of the linear functions topics covered in the Grade 9 course, provide students with the opportunity to develop algebraic and graphical models in the context of realistic problems. The timelines provided are suggestions to guide teacher planning and can be modified to suit the needs of individual classrooms. Teachers may consider including locally relevant applications of the topics covered in these activities, where appropriate. The activities provided involve considerable group work, and thus provide the students with opportunities to demonstrate Catholic values as collaborative contributors and Christian leaders. Many of the activities also provide opportunities for students to demonstrate their understanding of responsible citizenship and societal awareness. Activity 1.1 The Peace and Development Fundraiser: Intersection of Linear 75 minutes Models [construction of linear graphs from tables of values: interpolation: intersection of lines: formula development and evaluation (word and algebraic equations)] Activity 1.2 75 minutes Bottled Water Dilemma [use of technology in graphing; intersection of lines; subtraction of equations] Follow-up Skills Finding the intersection of lines using graphing calculators or 75 minutes graphing software; solving contextual problems Activity 1.3 75 minutes The Calculator Workshop [algebraic method of elimination] Follow-up Skills Practice with the elimination method; introduce the substitution 255 minutes method; practice with realistic situations Activity 1.4 75 minutes Battle of the Bands: Introduction to Quadratics [construction of graphs from tables of data: maximization of revenue and area; graphing a product of binomials; quadratic regression; connect product of binomials with ax2 + bx + c] Follow-up Skills Expand and simplify second degree polynomial expressions 75 minutes (squaring and expanding binomials) Activity 1.5 75 minutes Summative Activity: Paper and Pencil Test Unit 1 - Page 1 Principles of Mathematics - Academic Prior Knowledge Required Number Sense and Algebra manipulate 1st- degree polynomial expressions to solve 1st- degree equations add and subtract polynomials; multiply a polynomial by a monomial; expand and simplify polynomial expressions involving one variable solve problems, using the strategy of algebraic modelling Relationships determine relationships between two variables by collecting and analysing data compare the graphs of linear and non-linear relations collect, organize, and analyse data using appropriate equipment and/or technology describe trends and relationships in data construct tables of values, graphs, and formulas from descriptions of realistic situations and from data collected experimentally use interpolation and extrapolation to gather information from a graph distinguish between linear and non-linear relations by calculating finite differences Analytic Geometry identify the properties of line segments (direction, positive/negative slope, parallelism, perpendicularity) calculate slope using various formulae identify slope as a constant rate of change graph lines by hand and using graphing technology determine the equation of a line given slope and y-intercept, slope and a point on the line, and two points on a line, in the form y = mx +b and Ax + By + C = 0 Unit Planning Notes Practise using spreadsheets and graphing calculators in the context of the activities presented. Prepare to diagnose prior learning skills throughout the unit. Skill development activities should be developed to meet the needs that arise. Some students may require extended or enrichment activities to challenge their learning. Teaching/Learning Strategies It is expected that direct, teacher-lead instruction will be integrated within the framework of the activities as required to facilitate student learning and success. Independent practice of new skills will be necessary throughout the course. It is recommended that students be assigned to groups with any special needs and strengths considered. Appropriate peer grouping to benefit those students requiring extra help is suggested. Students will be involved in a considerable amount of group work. It would be beneficial to take some time to review appropriate group work dynamics, sharing of work responsibilities, assigning group roles, etc. Students should be encouraged to take ownership and responsibility for their own learning. Appropriate opportunities for students to communicate solutions, ideas, concepts should be provided throughout the course. Unit 1 - Page 2 Principles of Mathematics - Academic Assessment and Evaluation It is recommended that students be involved in the development of some of the rubrics used in the assessment/evaluation process. Students should be encouraged to self-assess and to identify areas that need improvement. Sample generic rubrics for oral presentations, written reports, and various learning skills are provided in this profile. They may be adapted to suit the needs of your classroom. Rubrics should be used where appropriate to assess student work. They are particularly appropriate to assess the expectations under the thinking/inquiry/problem-solving, communication and application sections of the Achievement Chart. When rubrics are used in assessment, students should be provided with the specific rubric that will be used prior to completing the assigned task. Care should be taken in the design of traditional paper and pencil tests to ensure that a Level 4 performance can be demonstrated. This unit is assessed using a paper and pencil test. Learning skills should be assessed in conjunction with the academic skills for each activity. Resources Graphing calculators; graphing software NCTM Addenda Series: Example 3 and Activity 2 from “Algebra in a Technological World” The Mathematics Teacher, Vol. 88 #3, March 1995: “Gas Bill Mathematics” Lappan, Glenda, et al. Moving Straight Ahead Linear Relationships. Dale Seymour Publishing, 1988. NCTM. Activities for Active Learning and Teaching. Activity 1.1: The Peace and Development Fund Raiser - Intersection of Linear Models Time: 75 minutes Description By developing tables of values, sketching graphs, writing linear equations, and interpreting the point of intersection, students will analyse a situation with variations. In this activity, the student council of Holy Mary High School has decided to do a walk-a-thon as one of its many fund raising activities and are faced with a decision of an appropriate, consistent donation plan for the whole student body to follow. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a discerning believer formed in the Catholic faith community who develops attitudes and values founded on Catholic social teaching and acts to promote social responsibility, human solidarity and the common good; - a self directed, responsible, life long learner who demonstrates a confident and positive sense of self and the respect for the dignity and welfare of others; - a collaborative contributor who works effectively as an interdependent team member. Strand(s): Analytic Geometry Overall Expectations AGV.01D - model and solve problems involving the intersection of two straight lines. Unit 1 - Page 3 Principles of Mathematics - Academic Specific Expectations AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation. Planning Notes The teacher will need a class set of graphing calculators and graph paper. The teacher will take the time to discuss the particular charity their school usually supports and the responsibility of all citizens to reach out to others in need. The chaplaincy office or the religion department in your school would be a place to obtain more information on this topic. There is opportunity to extend this activity later in this unit by using the other fund raising suggestions given by the students and solving systems of linear equations by algebraic methods. Prior Knowledge Required Relationships use a graphing calculator or graphing software on a computer construct tables of values and graph linear relations construct formulas derived from descriptions of realistic situations describe the effect on the graph and the formula of varying the conditions of the situation they represent determine the values of a linear relation by formula and by interpolating or extrapolating from the graph Analytic Geometry determine the point of intersection of the two linear relations, by hand for simple examples, and using graphing calculators or graphing software for more complex examples; interpret the intersection point in the context of an application. Teaching/Learning Strategies Teacher Facilitation: The students will work in small groups of two or three. The teacher will help any group experiencing difficulties in Part A of the activity. Students should be able to complete Part B with very little help from the teacher. During Part B the teacher will be able to observe and encourage students to work efficiently within a group setting. The students should be prepared to hand in this report and/or to give a quick oral presentation using chart paper or an overhead sheet to illustrate their charts, graphs and equations. Student Activity The students will work in small groups of two or three to complete the following handout: The Peace and Development Fund Raiser - The Student Handout This year the student council of Holy Mary High School decided to encourage students to raise funds for the Peace and Development Fund. The student council decided to encourage the students to participate in a ten-kilometre walk-a-thon as one of the many fund raising activities the school will be involved in during the year. During an organizational meeting, Andrew the student council president, asked what would be a fair donation rate per kilometre to ask of the sponsors, assuming each walker must have a minimum of 5 sponsors on their pledge sheet. Andrew then stated that the students in his home room suggested 75 cents per kilometre. Beth, who is the representative of another class, stated that perhaps the students can ask for a $5.00 donation plus 25 cents per kilometre. Unit 1 - Page 4 Principles of Mathematics - Academic Part A 1. Make a table showing the amount of money which would be pledged under each plan if the students walk up to 10 kilometres. 2. Using different colours, graph each pledge plan on the same coordinate axes. 3. At what point do the lines intersect each other? Explain what this means in the context of this situation. 4. For each plan, write a formula that will help the volunteer calculate the amount of money one sponsor owes, given the distance the student completed. Write a formula in words first and then in algebraic form. 5. If the student completed 7.5 kilometres, how much would the sponsor owe under each plan? Explain how you calculated your answer using two different methods. 6. a) If the sponsor owes $6.00, how many kilometres would the student have walked under each plan? Explain how you found the distance using three different methods. b) If the sponsor owes $7.20, how many kilometres would the student have walked under each plan? Explain how you found the distance using two different methods. 7. Beth suggested a $5.00 donation and then 25 cents per kilometre. a) How is this $5.00 donation represented on the graph? b) If the rate of 25 cents per kilometer was changed to 50 cents per kilometre, how would this change the graph? Using a different colour draw the line on the same axes as Question 2. Don’t forget to show a table of values for this situation and state the formula. c) The line in part b) intersects the line representing Andrew’s plan. Explain what the point of intersection means in the context of this situation. 8. By changing the initial donation or the donation rate or both, find a new formula which will give an $18 donation if the student completes 7 kilometres. Illustrate this new plan on a graph. 9. Which pledge plan would you suggest Holy Mary High School use in this walk-a-thon? Give reasons for your choice. Part B 1. Your home room does not like the plans suggested so far. a) Describe two other plans which could be used to raise funds in the walk-a-thon. The only restriction student council has placed on the pledge forms is that no sponsor should pay more than $20.00 for the completed distance. b) Make tables of values and graphs to illustrate these two plans. c) Write a formula for each plan. d) Do the two lines intersect each other? Hopefully so! Where? Explain what the intersection point means. e) Pick a distance before the intersection point and describe which plan is better. Explain why. f) Pick a distance after the intersection point and describe which plan is better. Explain why. g) From these two plans, which one would you prefer and why? 2. Using your chosen plan, what would the minimum amount of pledges be, if 20% of 1500 students from Holy Mary High School finished the walk-a-thon. 3. What other activities could this school use to raise money for the Peace and Development Fund? Assessment/Evaluation Techniques A variety of assessment tools should be used to properly evaluate the student: Through observation, make anecdotal comments on independent work, teamwork, organization skills, work habits, communication and initiative. A written report rubric (Appendix D) could be used on Part B, due to the variety of rate plans students would suggest. Unit 1 - Page 5 Principles of Mathematics - Academic If the teacher wishes to have presentations instead of a written report, verbal presentation rubric (Appendix C) could be used. A paper and pencil quiz would be used to assess whether the student can set up a table, graph, find the intersection point and explain what it means in the context of another realistic situation. Accommodations The teacher should be available to help students who are experiencing difficulties in completing Part A of this activity. In Part B the teacher can encourage students, reword the question if not clearly understood, and provide examples from Part A as a guide. Resources Lappan, Glenda, et al. Moving Straight Ahead Linear Relationships. Dale Seymour Publications, 1998. Activity 1.2: Bottled Water Dilemma Time: 75 minutes Description This activity provides students with a context for exploring graphical representations of a linear system of equations in two variables and interpreting the point of intersection of two linear relations. Using graphs they will examine and compare two different reward structures, for supply and sale of bottled water to a school, offered by competing companies. They will be looking for information and results to help them make a recommendation regarding which company will better serve the school’s bottled water consumption and fund-raising needs. The introduction of a pricing war provides an interesting extension which allows students to interpret the subtraction of equations in a graphical way and provides a good lead into the algebraic solution of linear systems. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a collaborative contributor who works effectively as an interdependent team member. Strand(s): Analytic Geometry Overall Expectations AGV.01D - model and solve problems involving the intersection of two straight lines. Specific Expectations AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation; AG1.03D - 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. Planning Notes The use of graphing tools, either graphing calculators or graphing software, is recommended since these allow students to apply and quickly see the effects of changes made to the reward schemes. They also enable students to easily and accurately find the coordinates of the point of intersection of the two lines representing the reward schedules. It is, however, also possible to carry out the activity without technology, but it would be more time-consuming. Unit 1 - Page 6 Principles of Mathematics - Academic This activity would be more meaningful for the students if they were able to use real data. Perhaps your school is considering the installation of a vending machine. Your Student Council might be pleased to recruit the help of Grade 10 students in the decision-making process. This activity could provide an opportunity to discuss some of the ethical issues around the use and abuse of water in the world. Prior Knowledge Required Number Sense and Algebra add and subtract polynomials and multiply a polynomial by a monomial; expand and simplify polynomial expressions involving one variable. Relationships graph lines by hand or using graphing calculators or graphing software. Teaching/Learning Strategies Student Activity Working in pairs, students will use graphing to investigate the two fundraising schedules in the following problem. Problem The Student Council must decide between two companies tendering to supply bottled water in a vending machine for the cafeteria. They want to make sure they get the “best deal.” On offer are the following Each month, Moose Country Water will pay the school 5 cents for every bottle sold after the first 1000 bottles. Northern Crystal Water will pay 7 cents a bottle after 2000 have been sold each month. Which company should the Student Council go with to raise as much money as possible? 1. Write a word equation and an algebraic equation to describe the relationship between funds raised and number of bottles sold under each company’s scheme. 2. Graph the equations and describe the graphs. 3. Use the graphs to compare the fundraising possibilities under each scheme. 4. Based on the graphs, what advice would you give to the Student Council about which company they should choose? Explain the significance of the point of intersection of the graphs for the two companies. 5. Write your recommendations to the Student Council in the form of a brief report. Include details of your graphical analysis. Now consider the following development: Moose Country Water, in an effort to secure the contract, offers an additional incentive of a $50/month donation to the school fund. On hearing about this, Northern Crystal immediately responds with a matching offer of a $50/month donation. How do these additional payments affect (a) the equations and graphs of funds raised vs. number of bottles? (b) your recommendations to the Student Council? Teacher Facilitation: If students are struggling with Questions 3 and 4, prompt them with probing questions like "Which company should they choose if, on average, 2500 bottles of water are bought from the machine each month? Which company should they choose if, on average, 5000 bottles are bought each month? It is recommended that students use estimation skills to develop an initial response. Unit 1 - Page 7 Principles of Mathematics - Academic Extension: Have students investigate then describe, graphically and algebraically, some strategies that each company could apply to improve their chances of winning the contract to supply water to the school. Teacher Facilitation: Students may need more support with this part. They should notice that both lines shift up by the same amount, but that where they intersect, the number of bottles sold is the same as before the incentive donations were added. Follow-up Skills: 75 minutes finding the intersection of lines using graphing calculators or graphing software; solving contextual problems Assessment/Evaluation Techniques Assessment in the Learning skills areas of independence and initiative is possible as students work on the activity. Individual student written reports will provide evidence of learning in all four Achievement Chart categories and could be evaluated using a rubric. Resources This activity is an adaptation of an example in “Algebra in a Technological World”, NCTM Addenda Series Activity 1.3: The Calculator Workshop Time: 75 minutes Description The analysis of production costs of two calculator models serves as a context for discussion of linear equations in the form Ax + By = C. Students will then carry out a graphical investigation which will lead to solution of equations by the elimination method. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands, and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a collaborative contributor who works effectively as an interdependent team member. Strand(s): Analytic Geometry Overall Expectation AGV.01D - model and solve problems involving the intersection of two straight lines. Specific Expectations AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation; AG1.02D - solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination; AG1.03D - 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. Unit 1 - Page 8 Principles of Mathematics - Academic Planning Notes Graphing calculators or graphing software may be used for this activity. Also, since students will be working with equations in the form Ax + By + C = 0, software such as Zap-a-Graph would be useful, especially for the exploration in the second part of the activity. Some work with graph paper and pencil should be encouraged. Plan to have students work in pairs so that they may discuss what they see as they progress through the exploration. Prior Knowledge Required Analytic Geometry graphing lines by hand (using a variety of techniques) and using technology Number Sense and Algebra solving first-degree equations, including equations with fractional coefficients, using an algebraic method Teaching/Learning Strategies Student Activity: An electronics company is producing two types of calculators, the PC1900 (the cheaper model) and the RC2000 (more expensive model). Two machines are required to make these calculators (Machine S and Machine T). The PC1900 requires 1 minute at machine S and 5 minutes at machine T. The RC2000 requires 6 minutes at machine S and 2 minutes at machine T. Machine S is available for 360 minutes per day and Machine T is available for 400 minutes per day (they are required for other purposes throughout the day). 1. Complete each chart: PC1900 RC2000 Number of Time on Time on Number of Time on Time on Calculators Machine S Machine T Calculators Machine S Machine T 1 1 2 2 3 3 4 4 . . . . . . x y 2. The bottom row of the above charts gives you the time for the day’s production. Use this information to complete the following chart: Time on Machine S Time on Machine T PC1900 RC2000 Total Time Available 3. Let x represent the number of PC1900s and y represent the number of RC2000s. State the production equations for each machine. It will be in the form Ax + By + C = 0. Machine S: ____________________ Machine T: ____________________ Unit 1 - Page 9 Principles of Mathematics - Academic 4. Graph the above equations (you may use graphing software). What do the points on each line represent? 5. At what point do these two lines intersect? What does this intersection point mean? Teacher Facilitation: When most students have completed this part of the activity, the teacher should take up the findings so far, as a whole class discussion. Emphasis should be placed on: the Ax + By + C = 0 form of the equation of a straight line. interpretation of the point of intersection as the solution that satisfies the necessary conditions for both Machine S and Machine T. Student Activity Note: This activity is intended to use technology! Do not graph by hand! Investigating Intersections In this activity you will be discovering an algebraic process to find the intersection point. Let’s first look at some simple cases: 1. 7x - y = 2 2x - y = -3 (I) Graph the above equations. (II) The coordinates of the point of intersection are: (III) Add together the two equations above. (IV) Graph the new equation on the same set of axes. (V) What do you notice about these three lines? 2. 8x + 5y = 1 3x + 2y = 1 (I) Graph the above equations. (II) The coordinates of the point of intersection are: (III) Add together the two equations above. (IV) Graph the new equation on the same set of axes. (V) What do you notice about these three lines? 3. x - 4y = 6 -x - y = 4 (I) Graph the above equations. (II) The coordinates of the point of intersection are: Unit 1 - Page 10 Principles of Mathematics - Academic (III) Add together the two equations above. (IV) Graph the new equation on the same set of axes. (V) What do you notice about these three lines? 4. In your own words, describe what is always true about the graph of the resulting equation when you add two equations together. 5. 2x + 3y = 6 (I) Graph the above equation. (II) Multiply each term of the equation by 5. Graph the new equation on the same axes. (III) Multiply each term of the equation by -4. Graph the new equation on the same axes. (III) Describe what you noticed about these graphs. 6. 2x + y = 3 x+y=1 (I) Graph these equations. (II) Graph + (III) Graph + 2 (IV) Graph + (-2) (V) Describe what you notice about each of these lines. (VI) Which lines, other than the original two, are the most important? 7. 3x - y = 2 x + 2y = 10 (I) (II) (III) (IV) (V) Graph the above equations. Graph + Graph 2 + Graph + (-3) Which lines, other than the original two are most important? Unit 1 - Page 11 Principles of Mathematics - Academic 8. Without graphing, find the point of intersection of the following pairs of lines: (I) 2x + 3y = 6 (II) x + 3y = -1 2x + y = -4 2x -y = 12 9. Recall the calculator problem you solved graphically: An electronics company is producing two types of calculators, the PC1900 (the cheaper model) and the RC2000 (more expensive model). Two machines are required to make these calculators (Machine S and Machine T). The PC1900 requires 1 minute at machine S and 5 minutes at machine T. The RC2000 requires 6 minutes at machine S and 2 minutes at machine T. Machine S is available for 360 minutes per day and Machine T is available for 400 minutes per day (they are required for other purposes throughout the day). Now use the algebraic method to find the numbers of each type of calculator you can produce in a day. 10. Revisit the Bottled water problem The Student Council must decide between two companies tendering to supply bottled water in a vending machine for the cafeteria. On offer are the following: Each month, Moose Country Water will pay the school 5 cents for every bottle sold after the first 1000 bottles. Northern Crystal Water will pay 7 cents a bottle after 2000 have been sold each month. Use an algebraic method to determine the point at which the amount of money raised would be the same for both companies. Teacher Facilitation: Initially students will need help with the concept and process of adding two equations. Extension Ask students to investigate the number of solutions to a linear system. What are the conditions for a system to have no solution? one solution? an infinite number of solutions? An investigation of linear programming techniques would provide another extension. Follow up-Skills: 255 minutes Students will need to practise this algebraic method of solution by elimination. The teacher should provide examples of systems for students to solve, as well as more problems in which it is necessary to formulate the system first. Following practise with elimination, students should solve systems by the algebraic method of substitution. Again, realistic applications should be provided. Assessment/Evaluation Techniques A paper and pencil quiz which contains some questions of the type in Question 8 above and a communication component in which students describe the algebraic method of solution they have discovered in the exploration. Accommodations Pair students with reading or writing difficulties with other students who will be able to help them. Resources The above exploration is an adaptation of an activity in “Activities for Active Learning and Teaching” NCTM. Unit 1 - Page 12 Principles of Mathematics - Academic Activity 1.4: Battle of the Bands – Introduction to Quadratics Time: 75 minutes Description In this activity, students will multiply linear expressions to obtain a quadratic expression. Students will also use the properties of a quadratic relationship to solve optimization problems. They will use the regression menu from the graphing calculator to find the curve of best fit and find that the product of binomials is also an expression in the form of ax2 + bx + c. The follow up to this activity will be expanding and simplifying polynomial expressions. The area high schools have decided to continue a band competition which was started last year. The High School Band Promoter has decided to change the ticket prices in order to maximize his revenue needed to pay for the advertising display in a mall close to the auditorium which he has rented for this event. He sets up spreadsheets to determine the best price to charge and the maximum floor space he can use at the mall. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a collaborative contributor who works effectively as an interdependent team member; - a responsible citizen who accepts accountability for one’s own actions. Strand(s): Quadratic Functions Overall Expectations QFV.04D - solve problems involving quadratic functions. Specific Expectations QF3.02D - fit the equation of a quadratic function to a scatter plot, using an informal process and compare the results with the equation of a curve of best fit produced by using graphing calculators or graphing software; QF4.02D - determine the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software. Planning Notes This activity should be done in pairs so students can discuss the problem and share ideas; however they should present an individual report with the aid of computer programs and/or graphing calculators. Graph paper is needed for this activity. Prior Knowledge Required Relationships How to use a graphing calculator or graphing program on a computer. Construct tables of values and graph a nonlinear relation derived from descriptions of realistic situations. Unit 1 - Page 13 Principles of Mathematics - Academic Teaching/Learning Strategies Teacher Facilitation: Introduce the situation about finding the proper price to charge or amount of space to be used in mall as not a random choice. Point out that there is a considerable amount of research and mathematics which is used to decide on proper pricing or the amount of floor space allotted to any display. A guest speaker from some retail consultant firm would be helpful in talking about this to students and allowing them time to investigate possible careers and jobs in this area. After some meaningful discussion, the activity can be presented to small groups of two or three students. Make the students aware that they will present an individual report. Student Activity Battle of the Bands – Student Handout 1. Doug, who is the promoter of this event has rented an auditorium to allow high school bands an opportunity to show their talent. Usually this event attracts 1000 people at $15 for each person. At this price, all the tickets were sold last year. Doug decided to set up a display at a local mall to encourage students to participate and register their bands. To pay for this display Doug was forced to increase the price of the ticket. In order to help him decide what would be a fair price to charge, he conducted a business survey and found that the number of tickets sold will decrease by 50 for every dollar increase. a) Form a hypothesis about the best price to charge. b) Complete the chart in your report. Number of Increases Ticket Price Number of Tickets Sold Revenue 0 1 2 3 4 . . . x c) State the revenue equation. Revenue = ( )( ) d) Use a graphing program and/or graphing calculator to create a graph of Revenue vs. Number of the Increases. Include the graphs in your report. Describe the shape of the graph in words. e) What price should Doug charge for a ticket to maximize the revenue? Explain how you found the best price. f) How many people would he expect to be in the auditorium with the new ticket price? g) What is the maximum revenue? Explain how this can be found using the graph. h) How much profit was made by the new price arrangement? i) Enter the data from the chart in part b) L1 as Number of Increases and L2 as Revenue to produce a scatter plot on the calculator. Using the regression menu on your calculator find the equation of the curve of best fit through the points. j) Compare the two equations from part c) and part i). Use your graphing calculator to input the two equations. Do they represent the same curve? Explain why. 2. The local mall has given Doug a rope of 8 metres for the perimeter of his display and will allow him to place the display in one of two places in the mall. In the north end of the mall, there is only one wall. So he would have to use the rope for the other three sides. The south end of the mall, has an area which consists of two walls and the rope would be used for two sides. a) Form a hypothesis about the best location and the best dimensions of the display area. Unit 1 - Page 14 Principles of Mathematics - Academic b) For each location set up a table with the following headings: Length of Display, Width of Display, Length of Rope, Area of Display. Include a line for the general case in your chart by letting x be the length of the display. Include at least three diagrams of the display for each mall location. c) State the area equation for each location. Area= ( )( ) d) For each location, construct a graph Area vs. Length of Display. e) Which location gives the maximum area for the display. Explain how you decided which location was the best. f) What is the maximum area? State the dimensions for the display. g) Enter the data to produce a scatter plot on the calculator. Using the regression menu on your calculator, find the equation of the curve of best fit for both curves. h) Input the two equations for the northern location on the calculator. Do they represent the same curve? i) Input the two equations for the southern location on the calculator. Do they represent the same curve? j) Why do two equations written in different form give the same curve? Follow-up Skills: 75 minutes QF1.01D - Expand and simplify second degree polynomial expressions Assessment/Evaluation Techniques Through observation, make anecdotal comments on independent work, teamwork, organizational skills, work habits, communication and initiative. A written report rubric (Appendix D) could be used to evaluate such areas as the clarity of communication and correctness of computation. Accommodations Place students having difficulties with written work or language with students who will assist them. Extra time may be given for students demonstrating difficulties in language skills. Activity 1.5: Summative Assessment Time: 75 minutes Description Students apply the skills they have developed during this unit to solve a problem about prices of two Driver’s Education courses that can be modelled by a system of linear equations. As well as the activity, students will write a paper and pencil test of skills. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands, and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a collaborative contributor who works effectively as an interdependent team member. Strand(s): Analytic Geometry Overall Expectation AGV.01D - model and solve problems involving the intersection of two straight lines. Unit 1 - Page 15 Principles of Mathematics - Academic Specific Expectations AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation; AG1.02D - solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination; AG1.03D - 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. Planning Notes The summative assessment for the unit should reflect the type of activities that were conducted during the unit. Thus, the assessment of this unit should include two parts: a realistic scenario that involves students working in pairs or groups to solve a linear system a paper and pencil test. A variety of application questions that require solving a linear system can be found in the textbook. As well, some of the suggested extensions in this unit could also be used as assessment activities. Prior Knowledge Required All the skills developed in this unit for using Linear Systems to solve problems Teaching/Learning Strategies Teacher Facilitation: Present the following problem to the class: The Safe-T-First Driver’s Ed Company offers two different packages of driver’s education courses: group classes or private lessons. The number of hours of classroom instruction and the number of hours of in-car driving instruction are the same for both groups, but the two packages are priced differently. Group Lessons Package: Student lessons cost $10 per classroom hour plus $6 per driving hour, for a total cost of $170. Individual Lessons Package: Private lessons cost $30 per classroom hour plus $20 per driving hour, for a total cost of $550. Do each of the courses offer enough in-car driving time to provide a safe and complete driving course? Provide a graphical solution to the problem, and support your answer using an algebraic solution. Student Activity Students work in pairs to discuss and solve the problem, completing individual solutions for submission at the end of the task. Teacher Facilitation: Circulate around the class and prompt students who are having difficulty. Some students may choose to gather data on a table to use for the graphical solution. Unless the students determine equations for each package, they will be unable to support their graphical solution algebraically. Students may need to be guided to use a system of two equations to determine the number of hours that are planned for classroom instruction and in-car instruction. Encourage students to use variables (h = number of classroom hours, d = driving hours) (Solution uses two equations 10h + 6d = 170 and 30h + 20d = 550 to give 5 classroom hours and 20 driving hours.) Unit 1 - Page 16 Principles of Mathematics - Academic Assessment/Evaluation Techniques Assess the student’s written report using the rubric for Assessing Written Reports (Appendix D) with the following criteria placed in the stated categories in the left column of the rubric: Assess Thinking/Inquiry and Problem Solving using Expectation AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation. Assess Written Communication using Expectation AG2.04 - communicate the solution to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions. Assess Application using Expectation AG1.03D - 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. Sample Questions For the Paper and Pencil Test 1. Below are three systems of equations: A) y = - 6x + 10 (B) 4x – 3y + 24 = 0 (C) 4x + 5y + 10 = 0 y= 5 x+7 2 2x + 5y + 50 = 0 y = 3x + 5 Do not solve these systems! State which method (elimination, substitution) would be most appropriate to solve each system. Explain why you chose each method. 2. (i) Develop a system that has: (a) exactly one solution; (b) no solution; (c) an infinite number of solutions. (ii) For each system you have developed in (i) describe a realistic situation that would lead to the system. Explain the meaning of the solution of each system in the context of the situation. 3. Two rental stores have a weekend special on posthole diggers; U-Dig-It store charges $50 for a damage deposit on its equipment plus $10 for each hour ABC Rental charges a $60 damage deposit and $9 for each hour. a) Using two different methods, find the number of hours for which the charge is the same? b) If you are using the equipment for only 6 hours, which rental store would be the best deal? Explain your choice. c) Which algebraic method do you use and why? 4. An electrical company charges a flat fee for a small electrical job, plus an hourly rate. A 2-hour job costs $76. A 6-hour job costs $130. What is the flat rate and the hourly rate charged by this company? Unit 1 - Page 17 Principles of Mathematics - Academic Unit 2: Analytic Geometry Time: 24 hours Unit Description Students will use analytic geometry to solve problems involving the properties of line segments and to verify geometric properties of triangles and quadrilaterals. Specific investigations will use these line segment properties to develop formulas for the lengths and midpoints of line segments; determine the equation of a circles centred at (0, 0); solve multi-step problems involving properties of line segments; determine the characteristics of triangles and quadrilaterals having fixed co-ordinates; investigate and verify geometric properties of triangles and quadrilaterals having fixed co-ordinates; and develop communication and problem-solving skills. Strand(s) and Expectations Ontario Catholic School Graduate Expectations: CGE2b, 2c, 3b, 3c, 3e, 4f, 5a, 5g, 7b. Strand(s): Analytic Geometry Overall Expectations: AGV.02, AGV.03. Specific Expectations: AG2.01D, AG2.02D, AG2.03D, AG2.04D, AG3.01D, AG3.02D, AG3.03D. Activity Titles (Time + Sequence) The following table provides a suggested sequence and timing for teaching this unit. The activities are designed to give students an opportunity to investigate the principles of analytic geometry and to apply these principles in a variety of ways. These activities employ a common theme involving the design and development of a piece of land into a new community. Students use this theme to apply various analytic geometry concepts while developing a portfolio for submission and presentation at the end of the unit. Each activity in the unit will help students to visualise the concepts of analytic geometry; to analyse situations involving these concepts; to develop an appreciation of the world around them; and to promote a respect for God’s creation and an understanding of the need to use resources wisely. Activity 2.1 150 minutes Delta Force Use the Lone Wind Place map to determine distances between various elements of the community. When available, The Geometer’s Sketchpad may be used to develop formulas for determining the lengths and midpoints of line segments. Use the Lone Wind Place map to develop a management line between the mid points of the cables connecting the cell towers Follow-up Skills Practise finding mid-points and lengths of line segments on a xy75 minutes plane using pencil and paper and a calculator Activity 2.2 75 minutes Circular Thinking Using the Lone Wind Place map, students analyse ambulance costs based on the distance from a fixed centre Follow-up Skills Practice with equations of circles centred at the origin on an xy75 minutes plane using pencil and paper Activity 2.3 150 minutes Watered Down Music Determine distance between towns for music reception purposes. Follow-up Skills Use right bisectors of a chord in the design of a sprinkler system 75 minutes Full period paper and pencil test Unit 2 - Page 1 Principles of Mathematics - Academic Activity 2.4 Follow-up Skills Activity 2.5 Follow-up Skills Activity 2.6 Follow-up Skills Activity 2.7 Freedom on the Beach! Use Lone Wind Place map and analytic geometry to determine the position of an object off the beach. Practise other applications of lengths and midpoints in triangles (e.g., altitudes, centroids, medians, etc.) Cell Power Use communication towers with parallelogram boundaries and transmitting power constraints to prove properties of a parallelogram Practice, consolidation, and application of skills and concepts of quadrilaterals (lengths, slopes and midpoints of sides, equations of diagonals, characteristics of segments joining midpoints, etc.) Oh Deer! Ticked! Explore relationships of line segments joining midpoints of the sides of a triangle Practise a variety of problems that will verify geometric properties of triangles or quadrilaterals with given vertex co-ordinates. (e.g., diagonals on a rectangle bisect each other; diagonals of a square are perpendicular, etc.) Portfolio Presentations Math Fair. Students display and present their final Lone Wind Place plan and reports for public viewing and assessment. Summative Assessment Paper and pencil test 110 minutes 115 minutes 75 minutes 150 minutes 75 minutes 150 minutes 90 minutes 75 minutes Prior Knowledge Required All expectations from the Grade 9 Number Sense and Algebra Strand (particularly NAV.03 and NAV.04). All expectations from the Grade 9 Measurement and Geometry strand (particularly MGV.03). The use of dynamic geometry software (e.g., The Geometer’s Sketchpad™) Unit Planning Notes Provide several copies of the Lone Wind Place Community Template for the various activities. Teachers may wish to enlarge the Template onto ledger size paper. For presentations, students may wish to use transparencies and overlays of their plans. The activities in this unit revolve around a central theme in which students design a new community. Components of this community will be cumulative, evolving as students proceed through the activities. The unit activities are designed to allow either a student-directed approach, a teacher-directed approach, or a mixture of both. With sufficient computer access, teachers may have students develop the skills and concepts of this unit in self-directed groups of three or four students. Teachers may wish to have student groups present interim reports of their progress at the end of activities. A final submission will include a completed map of the community, a planning report, and a class presentation. The final presentation of their plan and investigations will occur at the end of the unit with each group’s Culminating Assessment Package. Unit 2 - Page 2 Principles of Mathematics - Academic Students will address all items of the Analytic Geometry Strand while resolving many of the problems encountered by developers, planners, and/or contractors during the construction of a community (e.g., surveys, utilities, streets, bridges, buildings, churches, school, landscaping, ponds, lakes, rivers, recreation facilities, industry, etc.). There will be many opportunities to integrate topics from other disciplines (e.g., Economics, Social Sciences, Sciences, Ethics, etc.) and every attempt should be made to do so. There must be a balance of time spent between the investigative aspect of this unit and the paper and pencil approach. Timely and appropriate use of dynamic software and technology should be interwoven and integrated throughout. Teaching/Learning Strategies This unit provides opportunities for a balance of teacher-directed and student-directed activities while employing a variety of groupings (e.g., whole class, small groups, independent study, etc.) Students are provided with many of the design features of the Lone Wind Place community but teachers may encourage students to add their own features. A final report and presentation of their Lone Wind Place plan will be part of the summative assessment activity for the unit. Where appropriate, teachers may wish to have students maintain journals of their work in this unit. Dynamic software (e.g., The Geometer’s Sketchpad™) is used as an investigative tool, where possible, to develop and reinforce the various analytic approaches which meet the expectations in this unit. Throughout the activities in this unit, students could use the following model for problem solving: Hypothesise: Formulate hypotheses related to properties of geometric figures on the xy – plane within the context of the Lone Wind Place Community Explore/investigate: Through hands-on activities investigate their application in the Lone Wind Place map (e.g., The Geometer’s Sketchpad™, graph paper) Model/Formulate: Develop algebraic, graphical and/or tabular models Transform/Manipulate: Develop algebraic and graphical skills as needed in the context of their investigations Infer/Conclude: Re-evaluate their hypotheses in light of their learning and make inferences to extend their learning Communicate: Express their findings individually and in groups; orally and in writing. Assessment and Evaluation A balance of assessment tools and strategies is recommended. Activities that are used as teaching tools can and should be used as assessment tools. A list of recommended assessment tools are performance tasks, paper and pencil tasks (e.g., quizzes, diagnostic tests, worksheets, small assignments), written reports, oral presentations, observation, and peer assessment. The course overview lists a variety of assessment tools and strategies. Students should submit their drafts and preliminary results of their investigations at the end of selected activities. Interim feedback will give students an opportunity to make modifications for their final presentation. Resources The Geometer’s Sketchpad™ Satellite images of localities: http://www.terraserver.microsoft.com Unit 2 - Page 3 Principles of Mathematics - Academic Activity 2.1: Delta Force Time: 150 minutes Description With teacher direction, students will determine formulas for the distance between points, the lengths of line segments and the co-ordinates of the midpoint of any line segment. Where available, they may use The Geometer’s SketchPad™ to establish concepts and to develop formulas. Students will use the Lone Wind Place Community as a template for applying their models. They will begin a survey and layout of this tract of land for public use. On their survey they will need to establish locations for utility centres and to determine the distances between them. Students will include this as part of their Cumulative Assessment Package. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: an effective communicator who reads, understands, and uses written materials effectively; an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others; a reflective and creative thinker who creates, adapts, evaluates new ideas in light of the common good; a collaborative contributor who works effectively as an interdependent team member. Strand(s): Analytic Geometry Overall Expectations AGV.02 - solve problems involving the analytic geometry concepts of line segments. Specific Expectations AG2.01 - determine formulas for the midpoint and the length of a line segment and use these formulas to solve problems; AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions. Planning Notes Students will need a copy of the Lone Wind Place Template. Teachers will need a transparency of the Template. (Note: This Template will be used for subsequent activities.) Students need graph paper and a calculator. A computer lab may be used for the dynamic software investigation part of the activity. If this is the first time the students have used The Geometer’s Sketchpad™, they will need extra time to learn how to use the program. Prior Knowledge Required Basic Grade 9 skills with coordinate geometry; use of The Geometer’s Sketchpad™ and the Pythagorean theorem Teaching/Learning Strategies Teacher Facilitation: Hand out a copy of the Lone Wind Template. Present students with the project scenario. Establish groups of three or four. Class discusses the problem and intended activity. Unit 2 - Page 4 Principles of Mathematics - Academic Student Activity Introduction You have been awarded a huge contract to develop former Crown Land into a new Life Style Community. You need to plan the placement of community services such as fire halls, public utilities, distribution centres, communication towers, residential areas, zoning, etc. Main Street (east-west) and Bay Street (north-south) are the only roads that currently exist. The administrative centre (O) is located at the intersection of Bay and Main. The administration centre has been nicknamed O, in recognition of Orville Ridgin who formerly lived on the site. The site has been surveyed in one-kilometre sections. To assist with planning, your map has a grid in which one grid unit represents one kilometre. Initially, you will need to establish a street plan for future access to the whole region. You need to survey the land for roads and various critical sites necessary for the community Teacher Facilitation: Allow individual creativity. If the resources are available teachers may wish to preface this activity with a short class discussion on urban planning and the application of good design techniques involving mathematical and engineering principles, urban geography principles (e.g., Geography Profiles, an Urban Planner, etc.) and aesthetic features. Student Activity 1 Part A The head surveyor asks that you prepare a report which establishes a method for identifying sites and method for measuring distances between two different locations. Develop an equation model that will allow Community Administrators to find the distance between any two locations on the map. Teacher Facilitation: Using small groups of two or three students, discuss and hypothesize methods for determining the length of a line segment, given the co-ordinates of two points. Students may develop and test their hypotheses using calculators, spreadsheets and The Geometer’s Sketchpad™. Use The Geometer’s Sketchpad™ to develop a measurement model that determines the length of a line segment with one end at the origin. Generalize the model so it may be used to measure the length of any line segment. Action Computer Keystrokes 1. Display the axes Graph – Create Axes 2. Create a point C in quadrant 1 3. Draw a circle from the origin A(0, 0) to the point C 4. Draw line segment AC 5. Display co-ordinates of C and length of segment AC Measure - Co-ordinates Measure - Length 6. Draw line segment CD to intersect the x-axis Construct – Perpendicular Line perpendicularly 7. Display length of CD 8. Draw line segment AD Note: AC, CD, and AD are sides of a right-angled triangle, ACD 9. Display length of AD 10. Move point C and note relationship of coordinates and the measured lengths 11. Use the SketchPad calculator to create relationships to investigate and create a hypothesis Unit 2 - Page 5 Principles of Mathematics - Academic Part B 1. Extend your hypothesis by beginning at step 1 again, and having point A not at the origin 2. Establish the formula for the length of a line segment between any two points 3. Test the formula by repeating the above steps. Begin by creating a circle not centred at the origin 4. Create a table and present results to teacher and class Teacher Facilitation: While circulating, direct students in the following manner: Establish a grid reference pattern (e.g., (5E, 6N) which should ultimately be abstracted to (+5, +6)) Suggest that groups first consider locations in the NE quadrant (+, +) If necessary, hint about right-angled triangle properties that they learned in Grade 9. Have groups present their measurement model to the class for further discussion 2 2 Lead the students to a model similar to d x y Lone Wind Place Template Legend A, B, and C are Life Guard Stations (Activity 2.4). P, Q, R, and S are cellular towers (Activity 2.5). F, J, and H are the corners of the triangular Conservation Area (Activity 2.6). Unit 2 - Page 6 Principles of Mathematics - Academic Part B The head surveyor asks you to develop a preliminary survey for the placement of three cell phone towers. All need to be strategically placed in order to give maximum coverage between O, and the rest of the site. There is already an existing cell transmitter at the lighthouse (L). Each tower needs an underground backup line that will directly connect them to each other. For the following use the diagram shown below. In your report, justify your suggestions about your placement of the cell towers. Assume each tower has a range of 5 km and one at the lighthouse has a range of 7 km. Your report needs to determine the minimum amount of cable required to directly link the three towers to each other underground. The planning committee demands a complete report on how you calculated your answer. In addition, include a line from O to the tower closest to it, and another line from the lighthouse (L) to the tower closest to it. Once again, your results must be presented using formulas for distance between two points. Teacher Facilitation: Recommend that students position their towers at intersections of roads which themselves should lie along the grid lines. Recommend that their report consist of a table containing the coordinates of the towers and the distance calculations. If computers are available, their report may include a table developed on a spreadsheet. Have students present their reports and conclusions to the class. Part C Ideally, a cell phone will receive its signal from the nearest transmission tower. Each pair of transmission towers must be calibrated to achieve this balance. In order to calibrate the signals a technician will measure the signal strength of each pair of the cell towers at a point midway between them. You need to calculate and show the coordinates of the mid-points between each pair of the cell towers. The surveyor requires you to calculate the exact coordinates of these connection points to the nearest 0.1 kilometres and plot them on your template. Your report requires a table containing the coordinates of the towers, the coordinates of the midpoints between adjacent towers, and the transmission distances from each cell tower to the mid-point between each pair of towers. Be sure to include the towers at O and L in your calculations. Your results should be accurate to the nearest 0.1 km. Unit 2 - Page 7 Principles of Mathematics - Academic Teacher Facilitation: If dynamic geometry software is not available in the school preparation for the activity can be performed using graph paper and mathematical sets. The number of these constructions must be limited and adjusted according to time restrictions. Students should develop formulas for midpoints between each of the pairs of towers, the tower closest to lighthouse L and the tower closest to the administration centre O. A short lesson on determining the coordinates of the midpoints as an average of two values may be necessary. x x 2 y1 y2 M 1 , is the midpoint between P1 (x1, y1) and P2 (x2 ,y2 ) 2 2 Alternate Activity where Technology is Limited Use an individual computer with a projection unit for illustration purposes. Students should be allowed to do some or all of the demonstrations for their classmates. Use small groups of two or three students. Use The Geometer’s Sketchpad™ to determine a method for finding midpoints. Instructions for The Geometer’s Sketchpad™ (Only initial keystrokes are indicated) Actions Computer Keystrokes 1. Show Axes Graph - Create Axes 2. Draw Line from origin along x-axis to (5, 0) 3. Construct Midpoint Construct - Point at Midpoint 4. Label it Display - Show label 5. Select endpoint and label it 6. Display coordinates of endpoint and midpoint 7. Select endpoint and move. Observe values of coordinates 8. Repeat using a line from origin to (0, 5) 9. Repeat with line from O to (5, 5) 10. Hypothesise how to obtain midpoint, given O and endpoint co-ordinates 11. Move endpoint into other quadrants and describe what happens. 12. Be sure snap to grid is on 13. Fix one endpoint integer co-ordinates (e.g., 3, 3) 14. Move other endpoint to different co-ordinates and observe midpoint co-ordinates 15. Determine a model for determining midpoint coordinates 16. Test your model for all cases 17. Record and present your model for determining the midpoint Assessment/Evaluation Techniques The survey report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports until the end of the unit. Assessment Instruments Observe group and assess teamwork and initiative using the Learning Skills Rubric (Appendix A). Evaluate Lone Wind Place Planning Report using Written Report Rubric (Appendix D). Use paper and pencil tasks (e.g., quiz on finding midpoints and distance between points). Unit 2 - Page 8 Principles of Mathematics - Academic Follow-up Skills: 75 minutes Refer to the textbook for questions that calculate midpoints and the distance between points. Direct your students to investigate each of the following: Determine distances between origin and any point in any quadrant Find the distance between two horizontal points and two vertical points Determine the distance between two points anywhere on the Cartesian Plane Solve problems that determine the distance between any two points on a line Relate this to former studies about characteristics of a line and its slope. Activity 2.2: Circular Thinking Time: 75 minutes Description This activity enables students to develop the equation of a circle centred at the origin. They will develop their equations in small groups and communicate their results to the class. The initial problem will be presented with respect to the Lone Wind Place development project. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads understands and uses written materials effectively; - a collaborative contributor who works effectively as an interdependent team member. Strand(s): Analytic Geometry Overall Expectation AGV.02 - solve problems involving the analytic geometry concepts of line segments. Specific Expectations AG2.02 - determine the equation for a circle having centre (0, 0) and radius r, by applying the formula for the length of a line segment; identify the radius of a circle of centre (0, 0), given its equation; and write the equation, given the radius; AG2.04 - communicate the solution to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions. Planning Notes This activity continues using the Lone Wind Place development theme. A problem requiring the development and use of an equation of a circle will be presented to the class. The activity may be done using dynamic graphing software. Due to the relative shortness of this activity, the teacher should consider the expertise of students with the software, the availability of computers, and the time allocated for investigation and development of their report. Graph paper, a compass, and a calculator are required for each group of four students. Prior Knowledge Required Number Sense and Algebra Grade 9 Academic: use algebraic modelling to solve problems, specifically the Pythagorean theorem; communicate solutions using appropriate mathematical forms (e.g., tables, graphs) Unit 2 - Page 9 Principles of Mathematics - Academic Teaching/Learning Strategies Student Activity Students briefly review their progress with the Lone Wind Place land design project. Teacher Facilitation: Present the following communiqué from the Lone Wind Place Administration Office. Administration wishes to develop cost zones for their ambulance service. Zone A will end at 5 km from O and Zone B will be the area beyond Zone A. They want you to develop an equation that describes the boundary curve between these two zones so that the can correctly program their computer with the information. They wish you to report how you develop these equations in order to allow for future modifications. In large group discussion, spend 5-10 minutes reviewing and brainstorming possible hypotheses for solving this problem. Recommend that the problem be abstracted onto an xy-plane for exploration Break into groups of four and have students formulate a model Student Activity 1. On graph paper or The Geometer’s Sketchpad™, draw a circle centred at (0, 0) on the xy-plane and having a radius of 5 units 2. Select several points on the circle in the first quadrant. Use integer coordinates. 3. Create a table containing the coordinates of these points. Hypothesize a relationship between these points and (0, 0). 4. Continue investigations for points and ends of line segments in other quadrants. 5. Develop a report of your investigations. 6. Communicate their results individually and in groups, orally and in writing. Assessment/Evaluation Techniques The report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports until the end of the unit. Assessment Instruments Assess teamwork, independence, and initiative using the Learning Skills Rubric (Appendix A). Assess individual communication skills using Verbal Presentation Rubric (Appendix C). Use paper and pencil quizzes. Follow-up Skills: 75 minutes Practise with equations of circles centred at the origin on a xy-plane using pencil and paper Given centre and radius determine equation of a circle. Find coordinates of points on a circle using its equation. Find circle given coordinates of a point on the circumference. Unit 2 - Page 10 Principles of Mathematics - Academic Activity 2.3: Watered Down Music Time: 150 minutes Description Students will consolidate the concepts and skills addressed thus far in the unit (midpoints and lengths of line segments, and circles). These will be used in context with transmitting power of local radio stations (radii) and with the sprinkler system in a circular garden (chords and right bisectors). Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands, and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problem; - a collaborative contributor who works effectively as an interdependent team member. Strand(s): Analytic Geometry Overall Expectations AGV.02 - solve problems involving the analytic geometry concepts of line segments. Specific Expectations AG2.03 - solve multi-step problems, using the concepts of the slope, the length, and the midpoint of line segments; AG2.04 - communicate the solution to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions. Planning Notes Students should continue working in small groups (two or three students per group). Students will use a copy of the Lone Wind Place template. Each student will be responsible for their own written report. Students will need graph paper, mathematical set and calculators. Teaching/Learning Strategies Student Activity 1 1. Find the distance (as the crow flies) from the city centre (0, 0)) to: Paradise Island, Kloseton, Farfromton, Happyville, Fifthville 2. For what practical reason(s) may you want to know these distances? 3. There are three radio stations that transmit from the city centre: the local college station, COLG, that has a power to transmit up to 20 km; a Jazz station, CJAZ, that has a power to transmit up to 50 km; a rock station, CROK, that has a power to transmit up to 100 km. For a resident of each town, find what radio stations they will be able to receive. Unit 2 - Page 11 Principles of Mathematics - Academic Teacher Facilitation: Help student locate the five towns if necessary. To help the students with Question 2, the teacher may suggest that this new community has the most up to date medical facilities in the region. Hence emergency patients may have to be flown directly from the local towns to the city centre (0, 0). Student Activity 2 Surrounding the Administrative Buildings are several beautiful gardens manicured by the city’s parks and recreation department. The centre piece garden, named The Magical Circle, is a circular shaped garden with a radius of 50 m. A unique watering system is currently being used. There is a large soaker hose of length 71 m with each end anchored on the perimeter of the garden. A second soaker hose, of length 85 m, is anchored at the midpoint of the first soaker hose and runs perpendicular from the first soaker hose to the perimeter of the garden. Miray, a member of the city’s engineering department, has been asked to investigate this watering pattern and to suggest possible alternatives. In order to do this, she constructs a grid so that the centre of the garden is at the point (0, 0). She also locates all the anchor positions of the soaker hose and then heads back to her office. Unfortunately when she arrives, she cannot find all her measurements. All she can find is the anchor positions of the first soaker hose (A(40, 30) and B(-30, 40)). You are a young co-op student placed with Miray at the engineering department. She passes this information on to you to analyse. Unit 2 - Page 12 Principles of Mathematics - Academic Answer the following questions to help you out: 1. Find the equation of the circle. 2. Find the anchor positions of the second soaker hose. 3. Do you see any problems with this watering pattern? 4. What recommendations would you make to Miray? 5. Why do you think that this pattern was put in The Magical Circle gardens in the first place? 6. There is a main watering outlet at the centre of the garden. How close will the soaker hoses be to the watering outlet? 7. The line along which the second soaker hose is lying is the perpendicular bisector of the line along which the first hose is lying. Explain in your own words what the term perpendicular bisector means. 8. Based on your garden diagram, does the perpendicular bisector of a chord of a circle have any special properties? Give reasons for your answer. 9. Construct a circle of your choice. Draw in a chord and construct the perpendicular bisector of this chord. Does this perpendicular bisector have the same property you identified in Question 8? 10. Compare your results with the students around you. Does the property always seem to be true? 11. What conclusion can you make about a perpendicular bisector of any chord of a circle? Teacher Facilitation: While circulating about the room, ensure that the students have made an accurate diagram. If the students are having difficulty finding the anchor point on the perimeter of the garden for the second soaker hose, you may accept a graphical approximation. Assessment/Evaluation Techniques The report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports for their Culminating Assessment Package. Assessment Instruments Observe groups and assess ability to work independently, teamwork, work habits, organization, and initiative using the Learning Skills Rubric (Appendix A). Evaluate Report using Written Report Rubric (Appendix D). Evaluate oral presentation using Verbal Presentation Rubric (Appendix C). Assess appropriate criteria in the Observational Rubric (Appendix B). Follow-up Skills: 75 minutes Full-period Paper and Pencil Test (problems of the multi-step variety using the concepts of slopes, lengths, midpoints of line segments, and the equation of circles) Unit 2 - Page 13 Principles of Mathematics - Academic Activity 2.4: Freedom on the Beach! Time: 110 minutes Description Students will use a coordinate system (vertices) to find lengths, midpoints, slopes, and equations of lines that form triangles. These will be used to classify triangles and to find medians, altitudes, right bisectors (and their equations), along with circumcentre and centroid. The activity focuses on finding the circumcentre of a triangle formed by three lifeguard stations on the beach of our community. Strand(s) and Expectations Ontario Catholic Student Graduate Expectations The graduate is expected to be: - an effective communicator who reads understands and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills. Strand(s): Analytic Geometry Overall Expectations AGV.02 - solve problems involving the analytic geometry concepts of line segments; AGV.03 - verify geometry properties of triangles and quadrilaterals, using analytic geometry. Specific Expectations AG2.01 - determine formulas for the midpoint and the length of a line segment and use these formulas to solve problems; AG2.03 - solve multi-step problems, using the concepts of the slope, its length, and the midpoint of line segments; AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions; AG3.01 - determine characteristics of a triangle whose vertex coordinates are given; AG3.03 - verify geometric properties of a triangle or quadrilateral whose vertex co-ordinates are given. Planning Notes Students need copy of Lone Wind Place map. Students will need graph paper, mathematical set, and calculators. Investigate properties of triangular shapes (lengths of sides, altitudes, medians, right bisectors, circumcentres and centroids with and without grids). Consolidate basic definitions of all the above and for scalene, isosceles and equilateral triangles. Provide sufficient practice examples. If The Geometer’s Sketchpad™ is to be used reserve computer lab time. Modification of this activity is included where availability of technology is limited. Prior Knowledge Required classification of triangles according to length of line; segments; medians, altitudes, right bisectors, circumcentres, centroids; equations of lines; use of The Geometer’s Sketchpad™ if dynamic software is to be used Unit 2 - Page 14 Principles of Mathematics - Academic Teaching/Learning Strategies Teacher Facilitation: Pose the following problem. There are three lifeguard stations on the beach of the Lone Wind Place map (labelled A, B and C). The city has been given $500 000.00 from a local philanthropist that must be used to improve the appearance of the beach area. The city council, after consulting with a citizen’s group and a wellknown local artist, has decided to put a Freedom Fountain out in the water. For aesthetic purposes, the Freedom Fountain is to be put equidistant from each of the three lifeguard stations. How can they find this position? This is of particular interest because the fountain will have special lighting and hence expensive underground cables will be necessary. Teachers may preface this activity with a short class discussion (with journal entries) on possible relationships between three adjacent items in a city (e.g., city hall, fire station, and police station). A link can be made with the role of a city planner and the discipline of urban geography. The time spent on the investigative approach should be limited, with the majority of time spent on the analytic approach, for the following student activity. Student Activity 1. Plot at least six points on graph paper (label these points). 2. Draw at least two triangles using these points. 3. Find the lengths of the sides of the triangles. 4. Identify the various triangles according to lengths of sides (scalene, isosceles, and equilateral and put the corresponding definitions in your notebook). 5. What is an altitude? Draw the three altitudes for each triangle. 6. Find the equations of these altitudes. (Hint: the altitude is perpendicular to the base, so, if you know the slope of the base, you can use this to find the slope of the altitude.) 7. In a triangle, what is a median? Draw the three medians for each triangle. 8. Find the equations of the medians of the triangles. 9. Find the equations of the right bisectors of the sides of the triangles (put the corresponding definitions in your notebook). 10. Locate the circumcentre and centroid of each triangle on your drawing (put the corresponding definitions in your notebook). 11. Verify these coordinates using the equations you found in Questions 8 and 9. 12. Describe any other triangle centres you know about, and explain how you could determine their coordinates. Teacher Facilitation: Time spent on the above activity should be limited since most of these concepts have been addressed previously. While students are working on the activity the teacher should circulate about the room to aid and prompt students who are experiencing difficulties. Once students have finished the activity, the teachers may lead the class in a discussion to summarise the definitions and the observations in the above student activity (if there is a need). Return to the initial problem and brainstorm with the students how they can use the Lone Wind Place map (with grid) to solve the problem. Students then prepare a written report. The above Student Activity may be done using The Geometer’s Sketchpad™. Teachers may wish to demonstrate tools and other basic capabilities of Geometer’s SketchPad for the students or let them investigate and learn on their own, under the teacher’s guidance. This choice will depend on the extent that the students used this program in Grade 9 and to the extent that it was used in previous activities of this unit. Students should be encouraged do the analytic approach along with their investigation below (i.e., midpoints, lengths, equations). Return to the initial problem and brainstorm with the students how they can use the Lone Wind Place map (with grid) to solve the problem. Students then prepare a written report. Unit 2 - Page 15 Principles of Mathematics - Academic Alternate Activity Where Technology is Limited Group students to make maximum use of technology. Use computer and projection unit for demonstration purposes. Students should be allowed to do all of the demonstrations for their classmates. If dynamic geometry software is not available, the activity can be performed using graph paper and mathematical sets. The number of these constructions must be limited and adjusted according to time restrictions. Assessment/Evaluation Techniques The survey reports that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports for their Culminating Assessment Package. Assessment Instruments Assess knowledge and understanding by conferencing with students about their findings as they are working through the investigation using Verbal Presentation Rubric (Appendix C). Observe group and assess teamwork and initiative using the Learning Skills Rubric (Appendix A). Evaluate the report using Written Report Rubric (Appendix D). Follow-up Skills: 115 minutes Teachers should supplement this activity with textbook exercises (include a wide variety of paper and pencil type questions): equations of altitudes and medians lengths of altitudes and medians location of centroid Activity 2.5: Cell Power Time: 75 minutes Description Students will use a coordinate system (vertices) to find lengths and equations of lines that form quadrilaterals. These will be used to classify quadrilaterals and to find diagonals (and their equations) and their intersection points. The activity focusses on proving relationships involving the intersection of diagonals in a parallelogram. Strand(s) and Expectations Ontario Catholic Student Expectations The graduate is expected to be: - an effective communicator who reads understands and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a responsible citizen who accepts accountability for one’s own actions. Strand(s): Analytic Geometry Overall Expectations AGV.02 - solve problems involving the analytic geometry concepts of line segments; AGV.03 - verify geometry properties of triangles and quadrilaterals, using analytic geometry. Unit 2 - Page 16 Principles of Mathematics - Academic Specific Expectations AG2.03 - solve multi-step problems, using the concepts of the slope, its length, and the midpoint of line segments (e.g., determine the equation of the right bisector of a line segment, the co-ordinates of whose end point are given; determine the distance from a given point to a line whose equation is given; show that the centre of a given circle lies on the right bisector of a given chord); AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions; AG3.02 - determine characteristics of a quadrilateral whose vertex coordinates are given (e.g., the perimeter; the classification by side length; the properties of the diagonals; the classification of a quadrilateral as a square, a rectangle, or a parallelogram); AG3.03 - verify geometric properties of a triangle or quadrilateral whose vertex co-ordinates are given (e.g., the line joining the midpoints of two sides of a triangle is parallel to the third side; the diagonals of a rectangle bisect each other). Planning Notes Students will need Lone Wind Place map. Students will need graph paper, mathematical set, and calculator. Reserve computer lab time (if planning to use The Geometer’s Sketchpad to supplement the analytic approach to the activity). Modification of this activity is included where availability of technology is limited. Investigate properties of quadrilaterals (length and slope of sides, length and slope of diagonals). Consolidate basic definitions of all the above for squares, rectangles, and parallelograms. Provide sufficient practice examples. Prior Knowledge Required lengths, midpoints, slopes, equations of lines; classification of quadrilaterals; diagonals of quadrilaterals; use of The Geometer’s Sketchpad™ (if planning on using dynamic geometry software) Teaching/Learning Strategies Teacher Facilitation: Pose the following scenario. There are four communication transmission towers (P, Q, R, and S) indicated on the Lone Wind Place map (form a parallelogram). The city’s public works department is responsible for deciding on the strength of the transmitter that is needed for each tower. The choice for transmitters are: 1 km maximum distance; cost is $100 000 2 km maximum distance; cost is $175 000 5 km maximum distance; cost is $400 000 10 km maximum distance; cost is $750 000 If the cost to the community is to be kept to a minimum, what choices should be made for each tower? (Justify, with analytic geometry) Teachers may preface this activity with a class discussion (with journal entries) on possible relationships between four items in a city that would form some type of quadrilateral and possibly have an influence on such things as transportation routes and public utilities (water lines, gas lines, hydro, sewers, etc.) Teachers should lead the class in some example(s) of finding equations of line segments, lengths and midpoints of line segments for four vertices that can be used to form various quadrilateral shapes. The activity below can be done by the students with graph paper or with The Geometer’s Sketchpad™ or a combination of the two. If this activity is chosen then the time spent on it should be limited, with the majority of time spent on the analytic approach. If The Geometer’s Sketchpad™ investigation is chosen Unit 2 - Page 17 Principles of Mathematics - Academic teachers may wish to demonstrate tools and other basic capabilities of The Geometer’s Sketchpad™ for the students or let them investigate and learn on their own, under the teacher’s guidance. This choice will depend on the extent that the students used this program in Grade 9 and in previous activities in this unit. Student Activity 1. Put a grid (x- and y-axis) on graph paper (or The Geometer’s Sketchpad™ screen) 2. Plot several points that will allow the construction of several quadrilaterals. Try to create as many different types of quadrilateral as you can. 3. Find the lengths of the sides and diagonals of the quadrilaterals. 4. Identify the various quadrilaterals according to the lengths of their sides and measure of interior angles (square, rectangle, parallelogram). 5. Using the grid, find the slopes and equations of the various line segments. Make observations that indicate relationships between the line segments (sides and diagonals with respect to slope, lengths and points of intersection). 6. Compare your results with those found by other students. Make a summary of the properties of the line segments (including the diagonals) in each type of quadrilateral. Teacher Facilitation: While students are working on the activity above the teacher should circulate about the room to aid and prompt students who are experiencing difficulties. Once students have finished the activity, the teacher should lead the class in a discussion to summarize the definitions and observations in the above student activity. In particular, to help with the initial problem, students will need to have recognized that the diagonals in a parallelogram bisect each other. Return to the initial scenario and brainstorm with the students how they can use the community map (with grid) to solve the problem. The teacher should encourage (and prompt) students to find the midpoints of the sides of the parallelogram formed by the towers and the intersection of diagonals. Inform students about the “circular” transmitting patterns of the towers. (All points that are 1 km from the tower lie on a circle with radius 1 km. The transmitter will reach all points inside and on this circle.) The teacher may have to prompt students to find the length from the vertices to other vertices, to the midpoints of the sides and to the intersection point of the diagonals. The teacher may have to prompt students to recognize that the lengths found above will influence the choice for the size of the transmitter. Activity Where Technology is Limited Group students to make maximum use of technology. Use computer and overhead projection tablet for demonstration purposes. Students should be allowed to do some or all of the demonstrations for their classmates. If dynamic geometry software is not available, the activity can be performed using graph paper and mathematical sets. The number of these constructions must be limited and adjusted according to time restrictions. Paper folding can also be used to illustrate intersection and parallelism properties Assessment/Evaluation Techniques The report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports for their Culminating Assessment Package. Assessment Instruments Assess knowledge and understanding by conferencing with students about their findings as the students are working through the investigation using Verbal Presentation Rubric (Appendix C). Observe group and assess teamwork and initiative using the Learning Skills Rubric (Appendix A). Evaluate the report using Written Report Rubric (Appendix D). Unit 2 - Page 18 Principles of Mathematics - Academic Follow-up Skills: 150 minutes Teachers should supplement this activity with textbook exercises that include a wide variety of paper and pencil type questions dealing with quadrilaterals: equations of sides of quadrilaterals given the vertices and classification of quadrilaterals intersection of equations of lines that will yield vertices of quadrilaterals midpoints and lengths of sides of quadrilaterals length and equation of diagonals and line segments joining midpoints of sides properties relating to the above (e.g., diagonals of a rectangle meet at right angles, line segment joining the midpoint of the sides of a parallelogram is parallel to the other two sides and other multi-step type problems) Activity 2.6: Oh Deer! Ticked! Time: 75 minutes Description Students will investigate and prove the relationship that various line segments in a triangle have with respect to the sides of the triangle (e.g., a line segment joining the midpoints of any two sides of a triangle is parallel to the other side). They will use the triangular shaped Conservation Area on the Lone Wind Place Template. Strand(s) and Expectations Ontario Catholic Student Expectations The graduate is expected to be: - an effective communicator who reads understands and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a collaborative contributor who works effectively as an interdependent team member; - a collaborative contributor who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others; - a responsible citizen who accepts accountability for one’s own actions. Strand(s): Analytic Geometry Overall Expectations AGV.02 - solve problems involving the analytic geometry concepts of line segments; AGV.03 - verify geometric properties of triangles and quadrilaterals, using analytic geometry. Specific Expectations AG2.03 - solve multi-step problems, using the concepts of the slope, its length, and the midpoint of line segments; AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions; AG3.01 - determine characteristics of a triangle whose vertex co-ordinates are given; AG3.03 - verify geometric properties of a triangle or quadrilateral whose vertex co-ordinates are given. Planning Notes Students use a copy of the Lone Wind Place map template with the Conservation Area clearly labelled. Students will need graph paper, mathematical set, and calculator. Unit 2 - Page 19 Principles of Mathematics - Academic Reserve computer lab time (if using The Geometer’s Sketchpad™ as a resource). Students should work in pairs or small groups – classroom should be organized accordingly. The intent of this activity is to have students realize and prove (among other things) that a line segment joining the midpoints of two sides of a triangle will be parallel to the third side. Prior Knowledge Required lengths and midpoints of line segments slopes and equations of lines The Geometer’s Sketchpad™ (if planning on using dynamic geometry software) Teaching/Learning Strategies Teacher Facilitation: Before putting the students in pairs (or groups), pose the following situation. The Lone Wind Place Community has set aside a piece of land that has a wealth of natural habitat. It is called the LONE WIND PLACE Conservation Area. The area is determined by the points F, J, and H on the Community Map. There is a main entrance at point F, off Main Street. There is also another entrance at the marina at point J. The conservation area is completely enclosed with fencing joining the vertices F, J and H. There have been recent cases of Lyme disease in the region. The LONE WIND PLACE conservation area has a thriving deer population increasing the possibility of ticks which carry and transmit this disease. Julia, an ecology graduate student from the local university, has been given a summer job to tally the deer population in the conservation area. Because of the size of the area, Julia has decided to split the conservation area into four congruent (same area) triangular regions. This will allow her to apply sampling principles to estimate the total number of deer by finding the number of deer in any one of the areas. Julia has decided to choose her four congruent triangles by ensuring that each region has two of its boundaries on the fence lines of the conservation area (see accompanying diagram). Where should the boundaries of these four triangular regions be constructed to ensure that all four regions have the same area? (use analytic geometry to justify) Student Activity Part A Instructions (The Geometer’s Sketchpad™ can be used as a resource with this activity) 1. Find the conservation area (determined by points F, J, and H) on the Lone Wind Place map. 2. Plot these points on The Geometer’s Sketchpad™ screen (or on graph paper). 3. Construct the triangle determined by these points. 4. Choose 3 points (one on each of the sides of the conservation area). 5. Construct 3 line segments using the 3 points chosen in Question 4. 6. Find the area of the 4 triangular regions determined by the 3 line segments. 7. If these areas are not the same, move the positions of some or all of these 3 points until the area of the 4 triangular regions are the same. 8. Find the coordinates of the 3 points. Unit 2 - Page 20 Principles of Mathematics - Academic Teacher Facilitation: While circulating about the room, the teacher may have to prompt the students to choose the midpoints of the sides of the conservation area. Once the students have finished Part A, the teacher should have a short class discussion on the choice of these 3 points and what properties the line segments joining these 3 points have relative to the sides of the conservation area. Part B Instructions 1. Find the equation of the three sides of the LONE WIND PLACE Conservation Area. 2. Find the midpoints of the three sides of the LONE WIND PLACE Conservation Area. 3. Find the equation of the three line segments that join the pairs of midpoints of the sides of the LONE WIND PLACE Conservation Area. 4. What do you notice about these line segments compared with the sides of the LONE WIND PLACE Conservation Area? 5. Refer to your answer(s) from Question 4. Do you think that this will be true for any triangle? 6. By choosing a triangle of your choice, prove or disprove. Be sure to include a well-labeled diagram with the three vertices clearly labelled. 7. How would Julia use these regions to estimate the number of deer in the LONE WIND PLACE conservation area? 8. Do you think there could have been a better way for Julia to do her job? Explain Alternate Activity Where Technology is Limited Group students to make maximum use of technology. Use computers and projection unit for demonstration purposes. Students should be allowed to do some or all of the demonstration for their classmates. If dynamic software is not available, the activity can be performed using graph paper and mathematical sets. Assessment/Evaluation Techniques The report that students have developed in the activities may be presented for preliminary discussion of concepts and ideas. Each group of students will retain their reports for their Culminating Assessment Package. Assessment Instruments Assess knowledge and understanding by conferencing with students about their findings as the students are working through the investigation using Verbal Presentation Rubric (Appendix C). Observe group and assess teamwork and initiative using the Learning Skills Rubric (Appendix A). Evaluate the Report using Written Report Rubric (Appendix D). Follow-up Skills: 150 minutes Use other resources (e.g., textbook) to find a variety of problems that will verify geometric properties of triangles or quadrilaterals whose vertex co-ordinates are given such as: diagonals of a rectangle bisect each other diagonals of a square are perpendicular line segment joining the midpoints of two sides of a triangle is one half the length of the other side medians of a triangle intersect at a common point (centroid) finding area of triangles and quadrilaterals from vertices (find appropriate lengths of sides and heights) Unit 2 - Page 21 Principles of Mathematics - Academic Activity 2.7: Portfolio Presentation Time: 90 minutes Description Students present their final unit portfolios in the form of a math fair or class presentations. Each group will display their final proposals and justifications for the Lone Wind Place Community Project as required by the activities of the unit. The final portfolios will consist of displays and presentations involving charts, maps, proposals, and mathematical justifications for the Community Project. Strand(s) and Expectations Ontario Catholic Student Expectations The graduate is expected to be: - an effective communicator who reads understands and uses written materials effectively; - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others; - a reflective and creative thinker who creates, adapts, evaluates new ideas in light of the common good; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a reflective and creative thinker who adapts a holistic approach to life by integrating learning from various subject areas and experience; - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills; - a collaborative contributor who works effectively as an interdependent team member; - a collaborative contributor who achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others; - a responsible citizen who accepts accountability for one’s own actions. Overall Expectations AGV.02 - solve problems involving the analytic geometry concepts of line segments; AGV.03 - verify geometry properties of triangles and quadrilaterals, using analytic geometry. Specific Expectation All specific expectations for AGV.02 and AGV.03. Planning Notes Teachers will need to provide for the use of audio-visual equipment, computer projection equipment, computers, and display areas. An alternate presentation area and/or time might be arranged for the Math Fair displays and presentations Some groups may wish to present expert speakers as part of their presentation. Teachers may wish to acquire the assistance of individuals who have been assisted with science or technology fairs. Assessment instruments need to be prepared for the class, to assist with peer assessment. Unit 2 - Page 22 Principles of Mathematics - Academic Teaching/Learning Strategies Teacher Facilitation: Each activity of the Lone Wind Place Project Survey contains reports of student investigations, analysis and general findings. After all the activities are complete a final presentation will be used to assess the overall capabilities of the group. These presentations may be in the form of individual presentations of the work of each group using a variety of presentation tools or as a math fair scenario in which all groups are responsible for setting up displays and explaining their results to other members of the class and visitors. Presentation skills, along with mathematical understanding of the concepts and skills of this entire unit will be assessed. Student Activity Students assemble their reports and investigations from all of the previous activities in this unit and develop a presentation package. The final presentation report containing all the results of each activity (thematic approach) allows students to role-play and present their results as a consulting firm report for committee consideration. It also allows groups to present their reports as individual group presentations using charts, paper reports, software such as PowerPoint, and/or other conventional methods currently being used in business. An alternative approach allows each group time to create a Math Fair display. Their results would be displayed using typical display and presentation tools for science and/or math fair environments. Assessment/Evaluation Techniques Any and all of the Rubrics in the Appendix can be used for student assessment. A sample skeleton Peer/Self-Assessment Checklist is included with this activity, since peer and self-evaluation are essential components of this activity. However, teachers should create their own checklist elaborating on the various criteria listed. Math Fair Peer/Self-Assessment Checklist Needs Satisfactory Improvement DISPLAY Organization Accuracy Appeal/Creativity WRITTEN Organization Mathematical Language Flow VERBAL Preparation Clarity of Explanations Mathematical Language DEPTH OF ANALYSIS Place a mark (X) in the appropriate box Unit 2 - Page 23 Good Excellent Principles of Mathematics - Academic Summative Assessment Time: 75 minutes Paper and Pencil Test that would include all aspects of this unit and should include all categories (Knowledge and Understanding, Thinking and Inquiry, Problem Solving, Communication) of the Achievement Chart. Unit 2 - Page 24 Principles of Mathematics - Academic Unit 3: Modelling Quadratic Functions Time: 41 hours Unit Description This unit will introduce, explore, and apply the properties of quadratic functions. Students will collect, analyse, manipulate, and display data from primary and secondary sources to model quadratic relationships. Students will use graphing technology and paper and pencil tasks to explore the characteristics, equations, and graphs of quadratic functions. Realistic applications will be used to develop the quadratic model and its properties. Algebraic techniques of simplifying, factoring, and solving quadratic equations will be developed throughout the unit. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be an effective communicator who: - reads, understands, and uses written materials effectively; - presents information and ideas clearly and honestly and with sensitivity to others. The graduate is expected to be a reflective and creative thinker who: - thinks reflectively and creatively to evaluate situations and solve problems; - adopts a holistic approach to life by integrating learning from various subject areas and experience. The graduate is expected to be a self-directed, life-long learner who: - demonstrates flexibility and adaptability; - applies effective communication, decision-making, problem-solving, time and resource management skills. The graduate is expected to be a collaborative contributor who: - works effectively as an interdependent team member; - respects the rights, responsibilities and contributions of self and others; - achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others; The graduate is expected to be a responsible citizen who: - respects the environment and uses resources wisely; - contributes to the common good. Strand(s): Quadratic Functions, Analytic Geometry Overall Expectations: All those from the Quadratic Functions Strand, AGV.01. Specific Expectations: All those from the Quadratic Functions Strand, AG1.01. Activity Titles (Time + Sequence) The following table provides a suggested sequence for teaching the Modelling Quadratic Relations unit. After (or during) each activity, the skills to be developed are stated, as well as the suggested timing to work on skill proficiency. The activities in this unit are designed to help students visualize and analyse the use of quadratic relations in the world around them. Many activities incorporate and require the use of graphing calculators or graphing software to consolidate the properties of quadratic relations before dealing with the abstract and algebraic study of these relations. Unit 3 - Page 1 Principles of Mathematics - Academic 3.1 Follow-up Skills 3.2 3.3 3.4 Follow-up Skills 3.5 Follow-up Skills 3.6 3.7 Follow-up Skills 3.8 Follow-up Skills 3.9 3.10 Follow-up Skills 3.11 Follow-up Skills 3.12 Follow-up Skills 3.13 Winds of Change (An Introduction to Quadratics in realistic situations) Algebra Skills: expand and simplify second degree polynomials The Twelve Days of Christmas Quadratic or Not Braking Distance (Determining the Equation of a Parabola using finite differences) Determining the Equation of a Parabola Graphs on the Move (Transformations on the Parabola y = a(x - h)2 + k Practise to recognize specific types of transformations What Goes Up Must Come Down/Ramp Cart Quadratic Highs and Lows (Determining x- and y-intercepts, maximum and minimum values) Interpret Real and non-Real roots graphically Common Factor, Trinomial factors, Difference of Squares Graphing Quadratics in ax2 + bx + c form using the xintercepts (quadratics that can be factored) Solve quadratic equations graphically and by factoring Graphing Non-Factorable Quadratic Equations in ax2 + bx + c form using the x-intercepts Max/Min problems Max/Min problems with and without realistic contexts A Square Deal! (Graphing Quadratics in ax2 + bx + c form by changing to a(x + h)2 + k form) Completing the square to graph a parabola Solving Max/Min problems by completing the square Root of the Problem (Development and use of the Quadratic Formula) Solve quadratic equations using quadratic formula Interpret Real and non-Real roots, number of roots Summative Assessment Activity Paper and pencil test 150 minutes 150 minutes 75 minutes 150 minutes 75 minutes 75 minutes 225 minutes 75 minutes 150 minutes 75 minutes 225 minutes 75 minutes 75 minutes 75 minutes 150 minutes 150 minutes 75 minutes 150 minutes 75 minutes 60 minutes 150 minutes Prior Knowledge Required Graphing skills, equation solving skills, adding and subtracting polynomials, use of graphing calculators. Unit Planning Notes Graphing calculators should be available daily. Check availability of the computer lab if using graphing software. Algebra tiles are suggested in Activities 3.7 and 3.11. After each investigation, the teacher should ensure that the mathematics to be developed during the activity has been drawn out. Plan enough time at the end of each investigation to bring closure to the activity, and perhaps write a concluding note or journal entry. Unit 3 - Page 2 Principles of Mathematics - Academic Teaching/Learning Strategies This unit provides opportunities for a balance of whole-class, small group, and individual instruction through student-centered and teacher-directed activities. The unit uses rich contextual problems which engage students and provide them with opportunities to demonstrate achievement of the quadratic function strand expectations. It is expected that all students will have the availability of graphing calculators during classroom activities on a daily basis throughout this unit. Students will graph the quadratic relation using a variety of methods that include: table of values, paper and pencil graphing calculators or graphing software x-intercepts of factorable quadratic equations transformations to y = x2 common factored form of y = ax(x - s) + t to locate two points and deduce vertex method of completing the square The method of completing the square will be the final method discussed so that students can appreciate the properties of quadratic equations of various forms before using an algebraic method of converting y = ax 2 + bx + c to the form y = a(x - h)2 + k. This method will only be used in situations involving no fractions. It is suggested that the method of completing the square be used as a last resort. Factoring trinomials of the form ax2 + bx + c will be performed by the method of inspection. The method of decomposition to factor these polynomials is strongly discouraged. Sufficient practice of skills should be incorporated so that students are proficient with quadratic relations with and without the use of technology (graphing calculators and software). Assessment and Evaluation A balance of assessment tools and strategies is recommended. The suggested assessment at the end of this unit consists of a summative assessment activity and an individual paper and pencil skills test. A variety of other assessment techniques are implemented throughout the unit to assess learning skills and the four categories of the Achievement Chart. Rubrics are found in the Appendices and should be used when the students perform the open-ended activities. (In the Course Overview, a variety of assessment tools and strategies are listed.) Resources Zap-A-Graph (Ministry licensed software) Green Globs, Spectrum (now available free for the TI-83 Plus) Unit 3 - Page 3 Principles of Mathematics - Academic Activity 3.1: Winds of Change Time: 150 minutes Description This activity provides students with a context for studying both linear and quadratic relationships in a graphical manner. Students will interpret the meaning of the intersection point of two lines, representing distance-time relationships in an air travel example. By investigating the effect of a change in one of the parameters (wind speed) on the position of this intersection point, they will generate data points that lie on a parabolic curve. Using the curve constructed, students will interpret the meaning of the maximum point and the intercepts in the context of the distance-time problem. By changing a different parameter (cruising speed of the plane), students will compare and contrast the shapes of the resulting parabolas. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: -a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; -a collaborative contributor who works effectively as an interdependent team member. Strand(s): Analytic Geometry; Quadratic Functions Overall Expectations QFV.01D - solve quadratic equations; QFV.03D - determine through investigation, the basic properties of quadratic functions; QFV.04D - solve problems involving quadratic functions; AGV.01D - model and solve problems involving the intersection of two straight lines. Specific Expectations QF1.01D - expand and simplify second degree polynomials; QF3.02D - fit the equation of a quadratic function to a scatter plot, using an informal process (e.g., a process of trial and error on a graphing calculator), and compare the results with the equation of a curve of best fit produced by using graphing calculators or graphing software; QF4.02D - determine the zeros and the maximum or minimum values of a quadratic function from its graph, using graphing calculators or graphing software; AG1.01D - determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a real situation. Planning Notes Graphs in this activity will be completed by hand and using graphing software or graphing calculators. The graphing calculator viewscreen for overhead projection will be useful for class discussion. CBR devices will be used by pairs of students to model the problem presented in the activity. Prior Knowledge Required Relationships construct tables of values and graphs to represent non-linear relationships derived from realistic situations understanding of the relationship: v = d/t Unit 3 - Page 4 Principles of Mathematics - Academic Analytic Geometry graph lines by hand and using graphing calculators identify practical situations involving slope identify the properties of the slopes of line segments Teaching/Learning Strategies Student Activity The students will work in pairs or in small groups to complete the following. Part A Pose the following problem to your students: Use the CBR and your graphing calculator to produce a Distance-Time graph which is composed of two straight segments. Set the CBR to record distance for 4 seconds. Teacher Facilitation: It may be necessary to provide students with instructions for the proper use of the CBR and how to use the appropriate graphing calculator program to run the CBR. To lead into the next activity, ensure that there is at least one group that constructs a triangle-shaped graph (outward and return segments). Have students present their graphs to the class (construct on board and discuss). Spend some time discussing the variations in graphs and features of the graphs (i.e., what slope represents; what relationship is graphed; independent/dependent variables; etc.). Discuss real-life examples that may produce similar graphs. Part B Teacher Facilitation: Present the following scenario to the students (a worksheet outlining the problem may be useful): A small passenger plane, belonging to Air Mathematica, is taking the math teachers at your high school on a 4 hour flight (the flight will be straight out and straight back). The plane is capable of cruising at a speed of 300 km/h in still air. The plane has enough fuel for a 4 hour flight. On the day of the flight there is a 50 km/h wind that will be with the plane on the outward journey, and against the plane on the return journey. Discuss, with the students, the effect that the wind will have on the speed of the plane. When the wind is with the plane, the plane’s resultant speed will be 350 km/h. When the wind is against the plane, the plane’s resultant speed will be 250 km/h. The worksheet for this activity is provided below. Students should be advised that they must walk forward and then backward (as opposed to actually turning around) to produce the appropriate graph. CBR SIMULATION ACTIVITY WORKSHEET 1. (a) How will you walk to simulate the outward and return journey of the plane? (b) How will the velocity you walk in the outward portion of the journey compare to the velocity you walk in the return portion of the journey? 2. (a) What is the significance of the peak on the graph? (b) What was your maximum distance from the CBR? (c) How long did it take you to reach this point? 3. Describe the slope of both the outward and return portions of your journey. Teacher Facilitation: At this point, time should be spent practising the construction of graphs using the slope and a point on the line. The concept of slope = rise/run should be reinforced. Unit 3 - Page 5 Principles of Mathematics - Academic Part C Teacher Facilitation: Pose the following question to the class: What is the maximum distance the plane can travel on the outward portion of its journey and still have enough fuel for the return portion of the trip? Note: Ensure that students have an understanding that the line representing the return trip will have a negative slope (negative velocity) and must be graphed accordingly. It is recommended that this activity be completed in a teacher-led, whole class format. Student Activity Use the following directions to create a distance/time graph to represent the plane’s outward and return journeys: Outward Journey 1. (a) What does the slope of a distance-time graph represent? (b) What is the velocity of the outward journey? 2. What coordinates represent the starting point of the flight? 3. Graph the outward portion of the journey on a sheet of graph paper. Return Journey 1. Is the slope of the line representing the return trip positive or negative? 2. What is the velocity of the return trip? 3. What coordinates represent the end-point of the flight? 4. Graph the return portion of the journey on the same axes as the outward journey. Analysis and Discussion 1. (a) At what point did your two lines intersect? (b) What does this point mean in the context of the problem? 2. (a) How far did the plane travel before turning around? (b) At what time did the plane turn around? Part D Teacher Facilitation: Students will work in pairs (or small groups) to determine the effects of differing wind speeds on the journey of the plane. Four different wind speed scenarios should be assigned to each group. It is important to ensure that two outward flights occur with the wind (e.g., a 25 km/h wind is with the plane; a 60 km/h wind is with the plane), and 2 outward flights occur against the wind (e.g., a 25 km/h wind is against the plane; a 60 km/h wind is against the plane), in order to achieve a complete parabolic curve. Unreasonable wind speeds may be assigned to provide the opportunity to discuss practical decision-making scenarios (e.g., Would planes actually take off in _____ km/h winds?). While students are working in pairs, circulate to check student work for understanding and accuracy. Provide chart paper for students to record their time and distance data. The whole-class data will be graphed. It may be necessary to discuss how to estimate curves of best fit. Spend some time discussing the shape/properties of the curve of best fit. When the class data is graphed using the graphing calculators, it may be useful to go through the steps with the students using the viewscreen overhead projection unit. The purpose of using the quadratic regression function of the graphing calculator is not to obtain the equation, but only to obtain the curve of best fit. Unit 3 - Page 6 Principles of Mathematics - Academic Student Activity 1. Using the new wind speeds provided by your teacher, determine the distance the plane can travel before turning around, and the time at which it must turn around, for each wind speed given. Organize your information in an appropriate format (e.g., chart) 2. Record your time and distance data on the classroom chart provided. 3. Using the class data, construct a distance/time scatter plot on graph paper. Estimate a curve of best fit for the data points. 4. Use your graphing calculator to verify your curve of best fit. Enter the class data into lists to obtain a scatter plot and use a regression program (Quadratic Regression) to obtain a curve of best fit. Discussion of Curve: (a) Describe the shape of the curve produced by the class data. (b) Is this curve symmetrical? If so, what type of symmetry does it have? (c) What is the maximum point on your curve? What does this point represent? (d) Under what conditions can the plane travel the furthest before turning around? (e) What do the x-intercepts of the curve represent? (f) Your graphing calculator shows that this curve extends below the x-axis. What meaning, if any, do these sections of the curve have in the context of the problem? (g) What conditions would produce the coordinates close to the x-axis? Discuss the implications of these situations for air travel. Part E 1. Hypothesize what would happen to the shape of the curve if: a) the cruising speed of the plane is changed from 300 km/h to 400 km/h b) the cruising speed of the plane is changed from 300 km/h to 200 km/h 2. Using the assigned wind speeds you were given in Part D: a) construct new piecewise linear graphs to represent the outward and return portions of the plane’s journey (this is to be done by hand on graph paper); b) identify the coordinates of the point at which the plane must turn around. 3. Record your data on a class chart. Use the pooled class data to construct two new curves on the same axes as the graph constructed in Part D. Use different colours to represent different curves. 4. Use your graphing calculators and the quadratic regression function to verify your curves. 5. How are the curves constructed in this section the same as the graph constructed in Part D? How are these curves different? Follow-up Skills: 150 minutes QF1.01 - expand and simplify second-degree polynomial expressions (e.g., (x - 4) (5x + 2) - 3(2x - 8)2) Assessment/Evaluation Techniques Through observation, make anecdotal comments on independent work, teamwork, organization skills, work habits, communication and initiative (see Rubric provided in Appendix). Ask students to record their work on this activity in the form of journal entries. Alternatively, have students complete a written report upon completion. Evaluate this report using the written report rubric (Appendix D). Accommodations The teacher should be available to help students who are experiencing difficulties with constructing graphs. In some cases, students may use calculator-generated graphs rather than construct graphs by hand (it is possible to download calculator graphs and print them out to provide students with a hard copy of the graph). Student pairings should be made so as to encourage peer tutoring. Unit 3 - Page 7 Principles of Mathematics - Academic Resources Materials from Shell Centre, Department of Mathematics, University of Nottingham Appendices Appendix A – Learning Skills Rubric Appendix B – Observational Rubric Appendix D – Written Report Rubric Activity 3.2: The Twelve Days of Christmas Time: 75 minutes Description Students will illustrate the data obtained from a familiar holiday carol graphically and will discover the shape of a quadratic function. Students will determine which type of gift was the largest in number and which type of gift was the smallest in number. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands and uses written materials effectively; - a collaborative contributor who works effectively as an interdependent team member. Strand(s): Quadratic Functions Overall Expectations QFV.04D - solve problems by interpreting graphs of quadratic functions. Specific Expectations QF4.02D - determine the maximum value of a quadratic function from its graph, using graphing calculators or graphing software. Planning Notes Graphs in this activity will be completed by hand and using graphing software or graphing calculators Bring in Copies of “The Twelve Days of Christmas” and a Math Dictionary Prior Knowledge Required Relationships Construct tables of values and graphs to represent non-linear relationships derived from realistic situations Analytic Geometry Graph lines by hand and using graphing calculators Unit 3 - Page 8 Principles of Mathematics - Academic Teaching/Learning Strategies Teacher Facilitation: Read “The Twelve Days of Christmas” together and present the problem to the class. Students will work in pairs in order to help each other through the assignment. The teacher will circulate among the various groups to assist those needing some guidance. The regression menu on the graphing calculators will be used only to generate the equation of the curve of best fit. Further discussion of this calculator tool is not required at this time. Students should be encouraged to experiment with the regression functions on the calculators. Student Activity Students will complete the following questions. The Twelve Days of Christmas 1. After twelve days of receiving gifts, the woman decided to total up all of her gifts. She wanted to know the following a) How many of each type of gift did she receive? i) How many partridges? (Remember she received a partridge every day for twelve days!) ii) How many turtle doves? iii) How many french hens? Continue the total for all twelve days Present the information in a table. b) Draw a graph to illustrate the data. When entering the data in the graphing calculator, let “X” be the gift number. For example Partridge is gift type “1”,Turtle doves are gift type “2”. What type of relation do you have? (Linear or non-linear?) c) Which gifts did she receive the most of and the least of? How is this illustrated on the graph? d) How many gifts did she receive in total? e) Does this curve show a quadratic relationship? Explain. 2. a) To show her appreciation, the woman decided to return the same type of gift giving to her true love. On the first day she gave a gift “a”, on the second day she gave gift “a” and two gifts “b’s”, on the third day she gave gift “a”, two gifts “b’s” and three gifts “c’s” and so on. (There were 26 days of giving various gifts to her true love, who is a math teacher) Be creative in your choice of gifts. Choose math-related items for each day (e.g., “a” represents something such as an “abacus”, “b” could represent a “balance”). Present your data in a table and illustrate the data on a graph. b) Include a list of your mathematical type gifts and the dictionary meaning of each. c) State which type of gift was the most numerous. Where is it located on the curve? d) State which type of gift was the least in number. Where is it located on the curve? e) Using the regression menu on the calculator, state the equation of best fit. 3. Homework assignment: Suppose that as a class, you have decided to collect food as a class project for the whole month of October (31 days). For example, on the first day you will collect 1 jar of jam; on the second day you will collect 1 jar of jam and 2 tins of tuna, etc. Which donation would be the least in number and which would be the most. (Consider which items would be most needed by your local food bank and ensure that these are the maximum number of items collected). Include tables of values and graphs to support your answers. Assessment/Evaluation Techniques Assess this activity using the learning skills rubric (independent work, organizational skills, work habits, communication and initiative) and assess the homework assignment in the areas of knowledge/understanding and communication using the written report rubric found in the appendices. Unit 3 - Page 9 Principles of Mathematics - Academic Resources Mathematics Teacher December 1995 Volume 88 Number 9 Appendices Appendix A – Learning Skills Rubric Appendix D – Written Report Rubric Activity 3.3: Quadratic or Not Time: 150 minutes Description This activity will allow students to investigate various relationships and distinguish which relationships are quadratic. The use of finite differences will help students make the decision about whether a relationship is quadratic or not. Using the table of values generated by various situations, students will use graphing technology and the regression menu of a graphing calculator to establish the connection between the curve and the equation. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a self-directed, responsible, life-long learner who demonstrates flexibility and adaptability. Strand(s): Quadratic Functions Overall Expectations QFV.03D - determine through investigation, the basic properties of quadratic functions. Specific Expectations QF3.03D - describe the nature of change in a quadratic function, using finite differences in tables of values, and compare the nature of change in a quadratic function with the nature of change in a linear function. Planning Notes The use of graphing tools, either graphing calculators or graphing software, is recommended since it will allow students to quickly establish which relations are quadratic and which are not. Although only five situations have been given, many other types of relationships can be investigated in a similar fashion. With a larger number of relationships to investigate, students could be placed in groups and then present their charts, curves, and equations to the whole class. Together, the class can draw conclusions on which curves produce quadratic equations. Prior Knowledge Required Number Sense and Algebra demonstrate facility in operations with integers Relationships construct tables of values and graphs to represent non-linear and linear relations derived from descriptions of realistic situations identify, by calculating finite differences in its table of values, whether a relation is linear or nonlinear Unit 3 - Page 10 Principles of Mathematics - Academic Analytic Geometry graph lines by hand, using graphing calculators or graphing software Teaching/Learning Strategies Teacher Facilitation: Place students in pairs so that they can discuss the investigation and help each other through the various scenarios. Each student is to complete the handout individually in order to have a record of this activity. At the end of the activity students should be given a new situation to complete individually and hand in. Each student should be able to set up a table of values, state whether the relationship is quadratic or not by using first and second differences, and use technology to produce a graph of the situation and to generate the equation. Student Activity For each of the following examples, complete the table of values, calculate the finite differences, analyse the differences to hypothesize about the shape of the graph and graph the relations. For each example answer the following questions: a) Are the first differences the same or different? b) Are the second differences the same or different? c) Is the relation linear, or non-linear? d) Using the regression menu on the calculator, find the equation of the curve of best fit. State the equation. Student Handout (A) Pizza Prices 1. A local pizzeria charges a flat rate of $7.00 for a medium pizza with one topping and $0.95 for each additional topping. What is the cost of a pizza with six additional toppings? First hypothesize about the cost of six additional toppings. Give reasons for your hypothesis. 2. Complete the chart. Number of Toppings Cost of Pizza First Differences Second Differences 0 1 2 3 4 5 6 3. What is the cost of a pizza with six additional toppings? Was your hypothesis correct? If not, what error(s) occurred in your initial hypothesis? (B) Floor Tiles 1. How many dark tiles are in a square floor of 144 tiles? What is your hypothesis? Unit 3 - Page 11 Principles of Mathematics - Academic 2. Complete the chart Width of Square Floor Number of Dark Tiles First Differences Second Differences 3 4 5 . . . 12 What is the number of dark tiles in a square floor containing 144 tiles? 3. Was your hypothesis correct? If not, what error(s) occurred in your initial hypothesis? (C) Population Growth 1. A special millennium math bug doubles in population every day. The population of this new type of bug was 300 on February 20th. What is the population of this bug on the 15th and 25th of February? What is your hypothesis about the population on these dates? 2. Complete the chart. Date Number of Days Population First Differences Second Differences Feb. 15 . Feb. 20 0 Feb. 21 1 . 2 . . Feb. 25 5 3. What is the population of the millennium math bug on Feb. 15th? Feb. 25th? Was your hypothesis correct? If not, what error(s) occurred in your initial hypothesis? (D) Sum of Natural Numbers 1. What is the sum of the first nine natural numbers? What is your hypothesis? 2. Complete the chart. Number of Terms Sum First Differences Second differences 1 1 2 1+2= 3 1+2+3= 4 1+2+3+4= . . 9 3. What is the sum of the first nine natural numbers? Was your hypothesis correct? If not, what error(s) occurred in your initial hypothesis? Unit 3 - Page 12 Principles of Mathematics - Academic (E) The Pizza Function 1. If you make six straight cuts in a circular pizza, what is the maximum number of pieces you can have? What is your hypothesis? 2. Complete the chart. Number of Cuts Number of Pieces First Differences Second Differences 0 1 2 . . 6 What is the maximum number of pieces using six straight cuts? 3. Was your hypothesis correct? If not, what error(s) occurred in your initial hypothesis? (F) Summary Report: i) Submit all charts, graphs and questions completed in this activity. ii) State the equations of the three quadratic curves of best fit in this exercise. iii) What are the similarities in these 3 curves? What are the similarities in the corresponding equations for these 3 curves? iv) State how you would distinguish a quadratic from a linear relation from the difference columns. v) Invent an equation that would give you a quadratic curve. Test your equation using the graphing calculator. Using a table of values, show that you have a quadratic relation. Also draw a sketch of your curve. Assessment/Evaluation Techniques Individual written reports of the results of these investigations will provide evidence of achievement in the knowledge/understanding, thinking/inquiry/problem solving, and communication categories. Use the written report rubric (Appendix D). This written work would also be a valuable addition to a student portfolio. Assessment of independent work, teamwork, organizational skills, work habits, communication and initiative should also be made. A journal entry describing the special features of a quadratic relationship should be included. Resources Algebra in the Real World Appendices Appendix A – Learning Skills Rubric Appendix B – Observational Rubric Appendix D – Written Report Rubric Unit 3 - Page 13 Principles of Mathematics - Academic Activity 3.4: Braking Distance (Analysing data that is Quadratic to determine its equation, using finite differences) Time: 75 minutes Description Students analyse data that is quadratic using first and second differences to determine an equation for the relationship. They will analyse two situations: a) speed of a car and the braking distance required to stop the car at that speed and b) pollution in space. Students will determine an equation for each relationship and solve problems related to them. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a reflective and creative thinker who adopts a holistic approach to life by integrating learning from various subject areas and experience. - a responsible citizen who respects the environment and uses resources wisely. Strand(s): Quadratic Functions Overall Expectations QFV.03 - determine through investigation, the basic properties of quadratic functions; QFV.04 - solve problems involving quadratic functions. Specific Expectations QF3.02 - fit the equation of a quadratic function to a scatterplot using an informal process and compare the results with the equation of a curve of best fit produced by using graphing calculators or graphing software; QF4.03 - solve problems related to an application, given the graph or the formula of a quadratic function; QF3.03 - describe the nature of change of a quadratic function by using finite differences in the table of values, and compare the nature of change in a quadratic function with the nature of change in a linear function. Planning Notes Much of the lesson is teacher-directed, with students participating in discussion of teacher-posed questions and making calculations as required throughout the development of the lesson. Graphing calculators will be needed for determining the equation by regression. The overhead calculator screen will be helpful as well, to demonstrate the method of regression. Prior Knowledge Required Graphing non-linear data, calculating first and second differences from a table Teaching/Learning Strategies Teacher Facilitation: Because they are approaching age 16, students will be interested and involved in any discussion and applications to driving a car. In a full class setting, discuss the effect of speed on braking distance and the importance of knowing the distance required to safely stop a car. Much data has been gathered by various agencies studying road safety, and how road conditions and speed affect braking distance. Provide the students with the following data that relates speed and braking distance. (Note: Provide only the data in the left 2 columns at the beginning.) Have the students graph the scatterplot of the relationship using the graphing calculator’s table or, if technology is not available, Unit 3 - Page 14 Principles of Mathematics - Academic by hand. Next, calculate the difference columns. Students will determine that the data is quadratic from the 1st and 2nd differences. Sketch the shape of the continuous graph on paper or the board. Discuss why the left half of the parabola is often “missing” in realistic applications but occurs in abstract situations (due to the impossibility of negative values). (Note: Provide students with the data given in the left two columns only. The other columns are completed as a guide for the teacher, and should be developed together with the class) Table for Speed vs. Braking Distance Table for the standard quadratic y = ax2 + bx + c x y = ax2 + bx + c 1st Diff 2nd Diff. Speed (S) (B) 1st Diff. 2nd Diff km/h Braking Distance in m 0 0 0 c .6 a+b 10 0.6 1.2 1 a+b+c 2a 1.8 3a + b 20 2.4 1.2 2 4a + 2b + c 2a 3 5a + b 30 5.4 1.2 3 9a + 3b + c 2a 4.2 7a + b 40 9.6 4 16a + 4b + c 50 15 5 60 21.6 6 70 29.4 7 80 38.4 8 90 48.6 9 100 60 10 Answer these questions by extrapolating from the sketched graph. Start the right-hand table only when you begin to calculate data for the standard form quadratic. Student Activity (or class discussion) Examine the graph to answer the following questions: a) What is the stopping distance for a car speeding at 120 km/h? b) Police measure 12 m skid marks at an accident scene in a residential area. What was the speed of the car? Was the car being driven within the legal speed limit? c) A car travelling at 110 km/h on a highway brakes as a deer runs out from the ditch, 75 m in front of the car. Will the driver be able to stop the car in time to avoid hitting the deer? (This question could Unit 3 - Page 15 Principles of Mathematics - Academic be used as a formative assessment question to be completed at the conclusion of today’s activity. See Assessment/Evaluation Techniques near the end of this activity.) Teacher Facilitation: Discuss the difficulty of determining accurate answers to each of the questions above, building a need for a more accurate method to determine the answers. Students should express a need to find an equation for the relationship. Show them the following method for determining the equation of a quadratic relationship. To compare this quadratic with the standard quadratic equation, develop a table for the standard quadratic equation, y = ax2 + bx + c, for values of x such that 0 x 10. Student Activity Complete the table and then calculate both first and second differences. (The completed table will appear as shown on the right half of the table.) Teacher Facilitation: Ask the class to compare the two tables. Which value of the braking distance relationship can be determined when compared with the standard quadratic? They may need help equating 2a = 1.2 and solving to find that a = 0.6. Next, they should notice that 3a + b = 1.8. Substituting a = 0.6 into this equation yields 3(0.6) + b = 1.8 giving b = 0. Students will then note that a + b + c = 0.6, and after substitution 0.6 + 0 + c = 0.6, giving c = 0. Using the standard form of the quadratic y = ax2 + bx + c and the calculated values of a, b, and c; the equation for the braking distance quadratic will be y = 0.6x2. Student Activity Return to the three questions discussed earlier about skidmarks and the deer on the highway. Use the equation to determine the exact values for each question. Teacher Facilitation: While the students still have the data on a table of the graphing calculator, demonstrate the use of the regression capabilities of the calculator to determine the quadratic’s equation. Press STAT, CALC, and choose Quadratic Regression. Student Activity The equation determined by the calculator should be the same as the equation determined using finite differences. Clear the calculator in preparation for the next activity. Teacher Facilitation: Pose this problem: Debris floating in space is becoming a problem for space travel, and may some day pose a problem on earth. Floating in orbit around the earth are many unused and broken pieces of rockets, space probes and satellites, as well as large and small rocks and particles broken from man-made space objects as well as meteorites. The man-made space garbage is accumulating at a rapid rate and is becoming increasingly more dangerous. Space pollution is a serious issue. Even the smallest particle can cause great damage at the velocity it travels as it orbits the earth, and could cause death to astronauts in space vehicles. Scientists have approximated the following data about the amount of space junk orbiting the earth: n (number of years) J (junk in millions of kg) 1 (1990) 3.0 2 (1991) 3.8 3 (1992) 4.7 4 (1993) 5.7 Student Activity a) Determine if the relationship between the number of years and the amount of space junk is linear, quadratic, or neither. Support your answer. b) Determine an equation for the relationship of n and J using finite differences. Check your equation using quadratic regression on the graphing calculator. Unit 3 - Page 16 Principles of Mathematics - Academic c) Complete the table up to the year 2000, assuming the amount of space junk continues to follow the same pattern of increase. d) Ten years into the third millenium, how much space junk will be orbiting the earth? e) Compare the amount of space pollution when you graduate high school to the amount on your 40th birthday. f) Calculate an interesting (or frightening) fact about possible space pollution using your equation. Homework/Extension Activities: Determine the equations of the relationships examined during the previous days, using the method of finite differences. Check the equations using the regression capabilities of the calculator. Assessment/Evaluation Techniques Assess Knowledge using paper and pencil tasks. Assess Communication and Understanding using the sample problem about braking distance required to avoid hitting a deer. Students could solve the problem graphically as well as algebraically, comparing the two methods of solution and the effectiveness and accuracy of each method. Follow-up Skills: 75 minutes Determine the equation of a quadratic using the method of finite differences and the capabilities of the graphing calculator. Activity 3.5: Graphs on the Move Time: 225 minutes Description Students will investigate the effects of simple transformations on the graph of y = x2. Through these explorations, students will develop an understanding of the roles of a, h, and k in the graph of y = a(x - h)2 + k. While these investigations may be completed by hand, the process is facilitated by the use of graphing software and/or graphing calculators. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: -a collaborative contributor who works effectively as an interdependent team member; -a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems. Strand(s): Quadratic Functions Overall Expectation QFV.02D - determine through investigation, the relationships between the graphs and the equations of quadratic functions. Specific Expectations QF2.01D - 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; QF2.02D - explain the role of a, h, and k in the graph of y = a(x - h)2 + k. Planning Notes This activity is best carried out using Zap-a-Graph (or equivalent graphing software) and/or graphing calculators, but pencil and paper approaches may also be used. Unit 3 - Page 17 Principles of Mathematics - Academic Reserve computer lab time for two or three periods if graphing software is to be utilized. If students have not previously used graphing software, an initial workshop on the features and operation of the program will be necessary. Students may work in pairs to complete this activity. Pairing should be done with ability levels in mind. Prior Knowledge Required Relationships graph lines by hand or using graphing calculators or graphing software. Teaching/Learning Strategies Student Activity Working in pairs, students will investigate the effects of simple transformations on the graph of y = x2. These explorations may be completed by hand or using graphing software (Zap-a-Graph) or graphing calculators. Teacher Facilitation: If graphing technology is to be used, it may be necessary to conduct a preliminary workshop on the features and operation of the graphing software for students who have not previously used the program. Students may work in pairs to complete this investigation. On the Move Worksheet Part A: Comparing y = x2 and y = ax2 Using your graphing calculators (or Zap-a-Graph) construct the following graphs and sketch the resulting graphs on the axes provided. Choose a different colour for each graph for easier analysis: (i) y = x2 (ii) y = 2x2 (iii) y = ½x2 (iv) y = -2x2 (v) y = - ½x2 (a) What effect does changing the coefficient of x2 have on the graph? (b) Without using graphing technology, sketch what you think the graphs of: y = 3x2 and y = -3x2 will look like: (verify your hypothesis using graphing technology) Part A Summary Fill in the blanks to complete the following statements: In comparing the graphs of y = x2 and y = ax2: (I) If a >1: the graph of y = x2 is: _____ vertically by a factor of a. (II) If 0 < a < 1: the graph of y = x2 is: _____ vertically by a factor of a. (III) If a < 0: the graph of y = x2 opens: _____ . (IV) If a > 0: the graph of y = x2 opens: _____ . (IV) If a < -1: the graph of y = x2 is: _____ vertically. This graph would open: _____ . Unit 3 - Page 18 Principles of Mathematics - Academic (V) If –1 < a < 0: the graph of y = x2 is: _____ vertically and would open: _____ . Part B: Comparing y = x2 and y = x2 + k Using your graphing calculators (or Zap-a-Graph) construct the following graphs and sketch the resulting graphs on the axes provided. Choose a different color for each graph for easier analysis: (i) y = x2 (ii) y = x2 + 3 (iii) y = x2 - 3 (iv) y = x2 + ½ (v) y = x2 - ½ (a) What is the effect of adding a constant (k) on the graph of y = x2? (b) What is the effect of subtracting a constant (k) on the graph of y = x2? (c) Without using graphing technology, sketch what you think the graph of: y = x2 - 4 and y = x2 + 6 will look like. (Verify your hypothesis using graphing technology). Part B Summary Fill in the blanks to complete the following statements: (i) Compared with the graph of y = x2, the graph of y = x2 + k is: _____ . (ii) Compared with the graph of y = x2 , the graph of y = x2 - k is: _____ . (iii) This effect is called a(n): _____ . Part C: Comparing the graphs of y = x2 and y = (x ± h)2 Using your graphing calculators (or Zap-a-Graph) construct the following graphs and sketch the resulting graphs on the axes provided. Choose a different colour for each graph for easier analysis: (i) y = x2 (ii) y = (x - 5)2 (iii) y = (x + 5)2 (iv) y = (x + ½)2 (v) y = (x - ½)2 (a) What does the effect of adding a constant (h) have on the graph of y = x2? (b) What does the effect of subtracting a constant (h) have on the graph of y = x2? Unit 3 - Page 19 Principles of Mathematics - Academic (c) Without using graphing technology, sketch what you think the graphs of: y = (x + 6)2 and y = (x - 3)2 will look like. Verify your answer using graphing technology. Part C Summary Fill in the blanks to complete the following statements: (i) For y = (x - h)2, the graph of y = x2 moves h units: _____ . (ii) For y = (x + h)2, the graph of y = x2 moves h units: _____ . (iii) This effect is called a(n): _____ . Part D: Putting it all Together Consider the following: (i) y1 = 2x2 + 3 (iii) y1 = 3x2 + 4 (v) y1 = 5x2 - 1 2 2 y2 = -2x +3 y2 = 3x - 4 y2 = 1/5x2 + 1 2 2 (ii) y1 = 3(x + 2) (iv) y1 = 2(x + 3) (vi) y1 = 3(x + 1)2 y2 = 3(x - 1)2 y2 = -2(x - 3)2 y2 = 1/3(x + 1)2 For each case above, describe how the graph of y2 will differ from the graph of y1. Give reasons for your descriptions. Verify your answer using graphing technology. Consider the following: y = 3(x - 2)2 - 4 (i) Based on your observations in this investigation, describe how this graph will differ from the graph of y = x2. (ii) Consider the point (2, 4). Explain how this point is transformed from y = x2 to y = 3(x - 2)2 - 4. Part E: Identifying the Transformation: For each of the following graphs, describe which factors (a, h, and k) were effected to produce the transformation (from y = x2) shown: Follow-up Skills: 75 minutes Further paper and pencil practice to identify the roles of a, h, k in transformations; recognizing specific types of transformations Unit 3 - Page 20 Principles of Mathematics - Academic Assignment: Patterning with Parabolas Teacher Facilitation: Share the following design with the class and lead a discussion about how the graph may have been created (discussion how original curve may have been transformed to produce pattern shown): Student Activity Students will create their own parabolic designs by developing an equation in the form y = a(x - h)2 + k; and applying various transformations to it, using graphing software and/or graphing calculators: Student Worksheet 1. Develop an equation in the form: y = a(x - h)2 + k equation: 2. Apply any number of transformations to this graph by altering a, h, k values: transformations: 3. Graph the original equation and the transformations on the same set of axes using a graphing calculator or graphing software. Sketch your pattern on a sheet of graph paper. 4. Share your pattern with a classmate. Indicate to your classmate which graph was the original equation. Have your classmate try to determine the transformations that produced your pattern. Assessment/Evaluation Techniques Assessment in the Learning skills areas of independence and initiative is possible as students work on the activity. Student worksheets may be evaluated using a written work rubric, if desired. After paper and pencil practice, a quiz on key concepts of the section is recommended. Journal: Summarize the role of a, h, and k in the shape and position of a quadratic curve. Assignment 1: The point (2, 4) is on the curve y = x2; the point (2, 0) is on the curve y = a(x - h)2 + k. Find possible a, h, k values that could account for this. Assignment 2: Complete the assignment Patterning with Parabolas: This worksheet may be collected and evaluated using a written report rubric; attention should be given to creativity displayed in the pattern produced and to the understanding of the role of transformations in producing the pattern. Unit 3 - Page 21 Principles of Mathematics - Academic Resources Internet site: http://forum.swarthmore.edu/workshops/sum98/participants/sinclair/sample.htm (Investigating Functions Using Spreadsheets - allows students to make connections between equations, data ranges-differences, and graphs simultaneously) Green Globs: available for TI-83 Plus from TI website. Appendices Appendix A – Learning Skills Rubric Appendix D – Written Report Rubric Activity 3.6: What Goes Up Must Come Down/Ramp Cart Time: 150 minutes Description This activity will provide students with the opportunity to use the CBR, to collect experimental data, to illustrate the data graphically using a graphing calculator. The students will continue to investigate quadratic equations in the form y = a(x - h)2 + k as curves of best fit. They will observe the effect a has on the equation in an effort to find the equation which best represents their curve. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems. Strand(s): Quadratic Functions Overall Expectations QFV.03D - determine through investigation, the basic properties of quadratic functions. Specific Expectations QF3.01D - collect data that may be represented by quadratic functions, from secondary sources or from experiments, using appropriate equipment and technology. Planning Notes The teacher will need different types of bouncing balls CBR units, TI-83 graphing calculators, calculator to CBR linking cable a board to be use as a ramp (about 2 metres long) different types of carts (large enough for the CBR to sense) books to place under one end of the ramp Prior Knowledge Required Relationships collect data, using appropriate equipment and/or technology organize and analyse data, using appropriate techniques and technology Unit 3 - Page 22 Principles of Mathematics - Academic Teaching/Learning Strategies Teacher Facilitation: It is best to have students work in groups of three. Each group needs one person to give directions and record the data, another to work with the CBR and calculator and the third person to get the equipment needed and do the actual experiment. Have students read through the experiment first, discuss what role each will take and make an equipment list, before allowing the activity to proceed. The teacher will check the equipment list before allowing the students to proceed with the experiment. Warn students to read carefully and follow the instructions in order! Student Activity Part A: What Goes Up Must Come Down Follow and complete the handout. Instructions READ CAREFULLY and DO IN ORDER! 1. Connect the CBR to the calculator with the linking cable. 2. Turn on the calculator, press apps, select ranger program, press enter. 3. You will see prgmRANGER on the screen, press enter. Follow the instructions on the calculator until you see the MAIN MENU. 4. In the MAIN MENU, select 3: APPLICATIONS, press enter. 5. In the UNITS menu, select 1: METERS, press enter. 6. In the APPLICATIONS MENU, select 3: BALL BOUNCE, press enter and follow the instructions on the calculator. 7. Drop the ball at the same time you press the trigger button on the CBR. When the CBR stops recording, wait for the calculator to transfer the data to the graph. 8. Your graph should show at least four good bounces. If you are not pleased with your graph repeat the experiment by pressing enter and selecting 5: REPEAT SAMPLE. Try again! 9. When you are pleased with your ball bounces, sketch a Distance-Time relation. Include units. Collecting Data 1. You are going to select one parabola and eventually transfer this curve onto another set of axes for closer examination. Select the TRACE tool on the graphing calculator. Use the right cursor key to move the tracer to the bottom left point of the parabola. Record the coordinates of this point in the chart below. Continue using the right cursor key to find the maximum point (called the VERTEX). Record the coordinates of this point in the chart. Again using the cursor key, locate the bottom right point and record the coordinates of this point. Time (X) Distance (Y) Bottom Left Point (Xmin) Maximum Point (Vertex) (Ymax) Bottom Right Point (Xmax) Note: Ymin must be taken as the lowest value in the Y column. 2. Press ENTER. On the PLOT MENU, select 7:QUIT to exit out of the ranger program. 3. Press WINDOW key on the calculator. Input your data from the chart in #1. Use the cursor down key (not the enter key) to input the information! Xmin=__________ Xmax=__________ Xscl= 1 Ymin=___________ Ymax=___________ Yscl= 1 Xres= 1 When you are finished, press GRAPH key. Sketch a distance-time relation. Don’t forget the units on your grid. Unit 3 - Page 23 Principles of Mathematics - Academic Questions 1. Another form of the quadratic equation is y = a(x - h)2 + k. This form is called the vertex form of a quadratic. h is the X coordinate of the vertex and k is the Y coordinate of the vertex. Using your vertex and a = -1, enter the equation into Y=. Press GRAPH. Does this curve fit your parabola? If not, change the value for a until you have a good fit. 2. What is the quadratic equation which best describes your parabola? 3. You will find the equation for the second parabola by returning to the ranger program. Will this equation be the same as the first parabola? Explain why or why not. i) Press PRGM, select RANGER, press enter twice, follow the instructions on the calculator until you get to the MAIN MENU, select 4:PLOT MENU, select 1:DIST-TIME. ii) Repeat the instructions in Collecting Data. Time (X) Distance (Y) Bottom left point (Xmin) Maximum Point (Vertex) (Ymax) Bottom right point (Xmax) Note: Ymin must be taken as the lowest value in Y column. Input the following data in WINDOW Xmin=__________ Xmax=__________ Xscl= 1 Ymin=__________ Ymax=__________ Yscl= 1 Xres= 1 The equation of the second parabola [in the form y = a(x - h)2 + k]: _______________________ 4. Repeat for the third parabola. Time (X) Distance (Y) Bottom left point (Xmin) Maximum Point (Vertex) (Ymax) Bottom right point (Xmax) Note: Ymin must be taken as the lowest value in the Y column. The equation of the third parabola [in the form y = a(x - h)2 + k]: _______________________ 5. How long did it take to complete the first bounce? second bounce? third bounce? 6. How high was the first bounce? second bounce? third bounce? Part B: Ramp Car Experiment Instructions 1. Place about three textbooks under the one end of the ramp 2. Place the cart at the bottom of the ramp and do a few practice runs with the cart. Give the cart just enough push so it can reach the top of the ramp and come back down. 3. Place the CBR at the bottom of the ramp. Connect the CBR to the calculator using a linking cable. Run the Ranger Program on the calculator until you get to the MAIN MENU, select 1:SETUP/SAMPLE, press enter. Unit 3 - Page 24 Principles of Mathematics - Academic 4. Use the cursor keys and the enter key to change the information to match the illustration. MAIN MENU START NOW REALTIME: NO TIME (S): 3 DISPLAY: DIST BEGIN ON: [ENTER] SMOOTHING: LIGHT UNITS: METERS When you have finished changing the screen, move the cursor to START NOW. Press enter. Read the instructions on the screen. 5. Give the cart a push at the same time as you press enter. Be careful. Do not allow the cart to hit the CBR. Wait for the information to show up on the screen. 6. If you are not pleased with the graph, repeat the process by pressing enter, select 5: REPEAT SAMPLE from the PLOT MENU. 7. You want a well-shaped parabola! If there is extra data on the curve you will remove it by pressing enter. On the PLOT MENU, select 4: PLOT TOOLS and then select 1:SELECT DOMAIN 8. The calculator asks for “LEFT BOUND?”. Move the cursor to the lowest left hand point on the parabola, press enter. A vertical line will appear on the screen. Now the calculator asks for “RIGHT BOUND?”. Move the cursor to the lowest right hand point on the parabola, press enter. The calculator will now re-adjust the screen to accommodate the parabola you have chosen. Repeat steps 7 and 8 if you are not pleased with the curve. 9. Sketch this on a Distance-Time grid. Collecting Data 1. Use the cursor key to find the vertex of the parabola and the minimum points on your parabola. Record this information in the chart below. Time (X) Distance (Y) Bottom left Point (Xmin) Maximum Point (Vertex) (Ymax) Bottom right point (Xmax) Note: Ymin must be taken as the lowest value in Y column. 2. Press enter. On the PLOT MENU, select 7:QUIT to exit out of the ranger program. 3. Press WINDOW key on the calculator. Input your data from the chart above. Use the cursor down key (not the enter key) to input the information! Xmin=__________ Xmax=__________ Xscl= 1 Ymin=__________ Ymax=__________ Yscl= 1 Xres= 1 When you are finished, press the GRAPH key. Sketch the Distance-Time relation. Unit 3 - Page 25 Principles of Mathematics - Academic Questions 1. Using the vertex form of a quadratic [y = a(x - h)2 + k], input this equation onto the calculator. a is -1 h is ________________________________(X-coordinate of the vertex) k is________________________________(Y-coordinate of the vertex) Press GRAPH. Does this curve fit your parabola? If not, change the value for “a” until you have a good fit. 2. What is the quadratic equation which best describes your parabola? 3. a) How long did it take the cart to go up the ramp and finish at the same place it started? How would you find this on the graph? b) What is the furthest distance of the cart from the CBR? How would you find this on the graph? 4. a) If you placed the CBR at the top of the ramp instead of the bottom and repeated the experiment, would the graph look the same or different? Explain your answer. b) Sketch the graph. 5. Repeat the experiment by placing the CBR at the top of the ramp. Record the information in the chart The equation [in the form y = a(x - h)2 + k]: _____________________________________ Sketch the relation. Was your predicted curve correct? If not, what error(s) did you make? 6. Explain the role of a in the equation y = a (x - h)2 + k; 7. Given: y = 2x2 + 5x + 7: (a) Graph the equation (b) Find the vertex from the graph (c) Put the vertex coordinates (h, k) into the equation y = a(x - h)2 + k (d) Try different a values to make the new form of the equation fit the graph in part (a). Assessment/Evaluation Techniques Through observation, make anecdotal comments on teamwork, independent work, organization skills, work habits, communication, and initiative (see Rubric provided in appendix). Students will submit their lab report to be evaluated using the appropriate rubric. Resources Modeling Motion: High School Math Activities With The CBR. Texas Instruments Incorporated, 1997. Real-World Math with the CBL System. Texas Instruments Incorporated, 1999. Math and Science in Motion. Texas Instruments Incorporated, 1997. Appendices Appendix A – Learning Skills Rubric Appendix B – Observational Rubric Appendix D – Written Report Rubric Unit 3 - Page 26 Principles of Mathematics - Academic Activity 3.7: Quadratic Highs and Lows Time: 75 minutes Description Students will find the vertex, axis of symmetry, x- and y-intercepts, and maximum or minimum values using graphing calculators and some algebraic techniques. This activity will involve both teacherdirected and student-centered investigations of these critical characteristics listed. The concepts of intercepts and maximum or minimum values will be illustrated through realistic situations. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands, and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a collaborative contributor who works effectively as an interdependent team member; - a self-directed life-long learner who applies effective communication, decision-making, problemsolving, time and resource management skills. Strand(s): Quadratic Functions Overall Expectations QFV.01 - solve quadratic equations; QFV.04 - solve problems involving quadratic functions. Specific Expectations QF4.01 - determine zeros and the maximum or minimum value of a quadratic function, using algebraic techniques; QF4.02 - determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software; QF4.03 - solve problems related to an application, given the graph or the formula of a quadratic function; QF1.05 - interpret real and non-real roots of quadratic equations geometrically as x-intercepts of the graph of a quadratic function; QF1.02 - factor polynomial expressions involving common factors, differences of squares, and trimomials. Planning Notes graphing calculators and overhead projection calculator reserve computer lab if using graphing software copy of labsheet for each student steps to execute calculator specialty functions written on poster board or chart paper to facilitate students using calculator be familiar with the specialty functions and practise before assigning activity Prior Knowledge Required use of graphing calculator, definitions of x- and y-intercepts and ability to solve for intercepts algebraically, transformations of y = x2, graphing quadratics by hand Unit 3 - Page 27 Principles of Mathematics - Academic Teaching/Learning Strategies Teacher Facilitation: The student activity involves the use of the graphing calculator to locate the zeros (x-intercepts) and maximum/minimum values of quadratic relations. Lead the students through the first example, preferably using an overhead projection of the graphing calculator, to explain how to execute the required functions on the calculator. Students will then continue working in pairs or individually to complete the remainder of the examples. Student Activity: Quadratic Highs and Lows Individually or with a partner, use a graphing calculator, graph paper, and the chart below to investigate characteristics of quadratic relations. Graph each relation, one at a time, and complete the chart with all information before completing the next example. Sketch each relation using the information recorded. Quadratic Vertex Is the vertex a Max/Min y-intercept x-intercept(s) Equation max or min? value 2 a) y = x - 2x - 3 b) y = -x2 - 10x -21 c) y = 9 - x2 d) y = (x + 2)(x - 4) e) y = -(x + 3)2 - 5 Teacher Facilitation: Lead students through example (a) by asking specific questions and directing students on how to use the graphing calculator to verify answers. The following questions and directions are given as a suggestion to guide students through the first example. Note: The calculator instructions are given for the TI-83 Plus graphing calculator. Although examples have been chosen that yield integer values for x-intercepts and vertices, answers obtained on the graphing calculator using the CALCULATE functions may need to be rounded to the nearest integer value. 1. Graph y = x2 - 2x - 3 on the calculator. Make sure the dimensions of the window are set to properly view the quadratic. 2. Find and record the coordinates of the y-intercept. 3. Find coordinates of the x-intercept(s) from the graph and record. (Explain that an x-intercept is also called a zero of the quadratic function and a root of the quadratic equation.) 4. To verify the coordinates of the x-intercepts use the CALCULATE ZERO function on the calculator by doing the following: Press 2nd TRACE select 2:ZERO Use the left/right arrows and move cursor to a point located to the left of but near the desired zero (x-intercept) and press ENTER to set the left boundary. Move the cursor to a point located to the right of but near the same zero and press ENTER to set the right boundary. Press ENTER again and the coordinates of the x-intercept are displayed at bottom of screen. Repeat the procedure to read the second x-intercept if it exists. 5. Find and record the coordinates of the vertex. (Recall that the vertex is the ‘turning point’of the quadratic.) How does the location of the vertex compare to the location of all other points on the graph? Is it a maximum or minimum point? State the maximum or minimum value. (Emphasize that the maximum or minimum value of a function is the y-value or y-coordinate of the vertex and the vertex is the maximum or minimum point (i.e., a value is not a point)). 6. To verify the coordinates of the vertex use the CALCULATE MAXIMUM or MINIMUM function on the calculator as follows: Press 2nd TRACE select 3:MINIMUM (or 4:MAXIMUM) depending on the characteristic of the vertex. Unit 3 - Page 28 Principles of Mathematics - Academic Set the left and right boundaries around the vertex using the same method as for x-intercepts. Press ENTER and the coordinates of the minimum/maximum will be displayed at bottom of screen. 7. Draw a sketch of the function using the information recorded. Allow students to complete the remaining examples. Circulate about the classroom to aid students having difficulties working with the calculator or filling in the chart with the appropriate information. After students have completed the investigation, be sure to summarize the characteristics of the quadratic. It is suggested that an acetate is prepared with solutions of the chart and the sketch of each relation. Use the graphs to reinforce the following points: vertex is a maximum if graph opens down and a minimum if graph opens up vertex lies on the axis of symmetry vertex lies equidistant between x-intercepts if they exist depending on the form of the equation, certain information can be read directly: a) If y = ax2 + bx + c then the y-intercept is (0, c) b) If y = a(x - h)2 + k then vertex is (h, k) and axis of symmetry is x = h c) If y = (x - p)(x - q) then the locations of x-intercepts are (p, 0) and (q, 0) Practise finding the x and y-intercepts algebraically using examples (c) and (d) from the chart and then try example (b). Students should find that to solve for the y-intercept in example (b) they need to solve a quadratic equation. This will introduce the need to learn factoring of trinomials which will be addressed in the next follow-up lesson. Return to this question after practicing factoring in the follow -up skills lessons. The following activity, Dives Away!, is included to give meaning to the zeros and the max/min value in the context of a realistic situation. Students may work in pairs, individually or the teacher may use the activity for a class discussion, assign for homework or use as an assessment task. Student Activity: Dives Away! In a diving competition, a diver’s height, h, in metres, above the water is given approximately by the equation h = -t2 + 3t + 4 where t is the time in seconds after the diver leaves the diving board. Graph the relation using the graphing calculator and complete the following questions by viewing the graph. Teacher Facilitation: If this activity is used as an assessment task then delete the guiding questions that follow and give the following directions. “Use the graph and write a description of the diver’s path from the moment the diver leaves the diving board. Be specific by including mathematical terminology such as max/min values and intercepts and explain how these numbers relate to the diver’s path.” 1. What must be the minimum value for time? Explain. Set the window of the calculator to reflect your decision. 2. State the coordinates of the vertex. Verify using the CALCULATE MAXIMUM function on the calculator. What information does the vertex give about the diver? 3. Determine the h-intercept. Explain the significance of this value for the diver. Verify algebraically. 4. Determine the t-intercept(s). Verify using the CALCULATE ZERO function on the calculator. Explain the significance of this value with respect to the dive. 5. What is the exact location of the diver 5 seconds after leaving the diving board? Verify algebraically. (Substitute t = 5 into the equation and find h= -6 metres.) 6. Will the graph have a minimum point? Explain. (The graph of h = -t2 + 3t + 4 will only have a maximum point in the abstract sense however in relation to the diving situation the diver’s path will reach a minimum point at the time the diver changes direction to return to the water’s surface.) Unit 3 - Page 29 Principles of Mathematics - Academic Assessment/Evaluation Techniques The teacher can assess knowledge and understanding by conferencing with students about their findings as the students are working through the investigation. A paper and pencil skills quiz could also be administered to assess knowledge and understanding. A test on using the graphing calculator appropriately could be given to assess students’ ability to use technology. Communication can be assessed by having students prepare a solution to the Dives Away scenario through journal writing or by presenting a report of the solution to the class. Other realistic applications could be assigned in addition to the Dives Away example. Using rubrics found in the Appendices assess the written report for Knowledge and Application of determining zeros and the maximum or minimum values of a quadratic function both algebraically and by using graphing software as well as solving problems related to quadratics. Learning skills, specifically teamwork, independence, and initiative, can be assessed using the rubric found in Appendix A. Extensions Discussion of the axis of symmetry is not an expectation however it does enhance the study of the features of the quadratic function. A column could be added to the chart of the first activity to include the equation of the axis of symmetry. The teacher must define the axis of symmetry as a vertical line of reflection that divides the parabola into two mirror images. Recall that equations of vertical lines are of the form x = a number. The graphing calculator may also be used to insert the axis of symmetry by performing the following on the calculator: Determine the equation of the axis of symmetry by viewing the graph then verify the axis of symmetry using the CALCULATE VERTICAL function on the calculator as follows: Press CLEAR twice or until you have a blank screen. (Note: If the screen is not blank the following instructions will produce a vertical line through the current location of the cursor and not the typed integer value). Press 2nd PRGM select 4:VERTICAL and type in the integer value for the vertical line desired and press ENTER. The graph should contain a vertical line in addition to the parabola graphed. The vertical line should be the axis of symmetry if properly identified by viewing the graph. To delete vertical lines press 2nd PRGM select 1: CLRDRAW Follow-up Skills: 225 minutes QF1.05 - interpret real and non-real roots of quadratic equations geometrically as x-intercepts of the graph of a quadratic function. QF1.02 - factor polynomial expressions involving common factors, differences of squares and trinomials Provide sufficient practise on factoring and use only the method of inspection to factor trinomials of the form ax2 + bx +c. Avoid the method of decomposition. Factoring skills will be used in great detail to graph quadratics and solve quadratic equations in the unit activities that follow. Teacher Facilitation: Notes on Factoring Trinomials Using Algebra Tiles It is suggested that the concept of factoring trinomials of the form x2 + bx + c be developed concretely through the use of algebra tiles. Restrict all examples where b and c are positive integers until a pattern is developed, then extend the pattern to factor all trinomials without tiles. Unit 3 - Page 30 Principles of Mathematics - Academic Example 1: Create a rectangle with area of x2 + 5x + 6 with algebra tiles. Use one x2 tile, five x tiles, and six unit tiles. Arrange the tiles to create the rectangle shown and then determine the length and width of the rectangle. Make a concluding area statement for the diagram in the form Area = (length)(width). Length = (x + 3) Width = (x + 2) x2 + 5x + 6 = (x + 3)(x + 2) Example 2: Repeat the procedure from example one for a rectangle having area x2 + 7x + 10 Length = (x + 5) Width = (x + 2) x2 + 7x + 10 = (x + 5)(x + 2) Give several more examples watching that students have enough algebra tiles to create the rectangle then give the area of x2 + 15x + 56 where the number of tiles is insufficient to create the rectangle. Have students deternine a pattern by directing them to search for a pattern that relates the integers in the binomial factors to the values of b and c in x2 + bx + c. Summarize the general rule: Find two integers that yield a sum of b and a product of c for x2 + bx + c. Now use the pattern, not algebra tiles, to factor trinomials such as x2 - 3x - 28, x2 + 2x -35, etc. Resources Scully, J., B. Scully, and J. LeSage. Active Learning Series “Alge-Tiles”. Exclusive Educational Products. Unit 3 - Page 31 Principles of Mathematics - Academic Activity 3.8: Graphing Quadratics in ax2 + bx + c form using the x-intercepts Time: 75 minutes Description Students will graph quadratic equations of the form y = ax2 + bx + c by factoring to determine the xintercepts. They will then locate the x coordinate of the vertex midway between the x-intercepts, and calculate the y coordinate of the vertex by substituting the x coordinate into the equation of the quadratic and solving. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is: - a reflective and creative thinker who evaluates situations and solves problems; - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills. Strand(s): Quadratic Functions Overall Expectations QFV.02 - determine through investigation, the basic properties of quadratic functions; QFV.04 - solve problems involving quadratic functions. Specific Expectations QF2.04 - sketch, by hand, the graph of a quadratic function whose equation is given in the form y = ax2 + bx + c, using a suitable method (e.g., locate the x-intercepts if the equation is factorable; express in the form ax(x - s) + t to locate two points and deduce the vertex); QF4.01 - determine the zeros and the maximum and minimum value of a quadratic function, using algebraic techniques; QF1.03 - solve quadratic equations by factoring and by using graphing calculators or graphing software. Planning Notes Graphing calculators will be needed. The use of the calculator overhead display unit will enhance the visual part of the lesson and aid in the discussion determining vertex coordinates. Prior Knowledge Required Common factoring, trinomial factoring (x2 + bx + c as well as ax2 + bx + c), and factoring the difference of squares. Finding the zeros of a quadratic using the graphing calculator. Teaching/Learning Strategies Teacher Facilitation: The teacher will lead the class through an analysis of quadratics graphed using the graphing calculator. A full class discussion of each graph should lead to finding an algebraic method to graph the parabola if its quadratic equation can be factored. Student Activity Using the graphing calculator, graph y = 3x2 - 6x. Examine the graph to determine the x-intercepts, the equation of the axis of symmetry and the coordinates of the vertex. Examine the equation to determine an algebraic method to graph the parabola. Unit 3 - Page 32 Principles of Mathematics - Academic Teacher Facilitation: Lead the students to determine the x-intercepts by setting y equal to zero, common factoring the binomial and solving 0 = 3x(x - 2). Examining the line joining the x-intercepts (2, 0) and (0, 0) the students should see that the vertex occurs midway between the x-intercepts (i.e., where x = 1). Thus, the x coordinate of the vertex will be 1. Substituting x = 1 into the equation y = 3x2 - 6x the y coordinate is calculated as -3, giving the vertex point as (1,-3). Some further discussion: This graph passes through the origin. Will all quadratics of the form y = ax2 + bx have this property? (Yes) Discuss why this happens. Verify by graphing a few samples on the graphing calculator. Student Activity Graph y = x2 - 25. Examine the graph to determine the x-intercepts, the equation of the axis of symmetry and the coordinates of the vertex. Examine the equation to determine an algebraic method to find each of the values. Teacher Facilitation: Lead the students to determine the x-intercepts by factoring the difference of squares and solving 0 = (x - 5)(x + 5). Examining the line joining the x-intercepts (5, 0) and (-5, 0) the students should see that the vertex occurs midway between the x-intercepts (i.e., where x = 0). Thus, the x coordinate of the vertex will be 0. Substituting x = 0 into the equation y = x2 - 25 the y coordinate is calculated as -25, giving the vertex point as (0, -25). Some further discussion: This graph has its axis of symmetry on the y axis. Will all quadratics of the form y = ax2 + c have this property? (Yes) Discuss why. Verify by graphing a few samples on the graphing calculator. Student Activity Graph y = x2 + 2x - 8. Examine the graph to determine the x-intercepts, the equation of the axis of symmetry, and the coordinates of the vertex. Examine the equation to determine an algebraic method to find each of the values. Teacher Facilitation: Lead the students to determine the x-intercepts by factoring the trinomial and solving 0 = (x + 4)(x - 2). Examining the line joining the x-intercepts (-4, 0) and (2, 0) the students should see that the vertex occurs midway between the x-intercepts (where x = -1). Thus, the x coordinate of the vertex will be -1. Substituting x = -1 into the equation y = x2 + 2x - 8 the y coordinate is calculated as -9, giving the vertex point as (-1, -9). Some further discussion: This graph has two x-intercepts. Will all quadratics of the form y = ax2 + bx + c have two x-intercepts? (No) Discuss. Will all quadratics of the form y = ax2 + bx + c factor? (No) Students will examine this case in detail in future lessons. Student Activity Graph y = 2x2 - 5x - 3. Examine the graph to determine the x-intercepts, the equation of the axis of symmetry and the coordinates of the vertex. Examine the equation to determine an algebraic method to find each of the values. Teacher Facilitation: This quadratic requires more challenging factoring and a fractional solution. Lead the students to determine the x-intercepts by factoring the trinomial, setting y to equal zero, and then solving 0 = (2x + 1)(x - 3) to find intercepts at (-.5, 0) and (3, 0) and calculating the vertex at (1¼ , -1/8). Student Activity Practise graphing the following quadratics without the aid of a graphing calculator. Graph a) y = -2x2 - 6x (parabola opens downwards) b) y = x2 - 4x + 3 and c) y = 5x2 - 10x. Homework: Practise graphing quadratics that can be factored as in the examples studied today. Unit 3 - Page 33 Principles of Mathematics - Academic Follow-up Skills: 75 minutes QF1.03 - solve quadratic equations by factoring and by using graphing calculators or graphing software. Assessment/Evaluation Techniques Assess factoring skills informally as students work on the activity. Assess knowledge and understanding using paper and pencil tasks involving graphing without the use of a graphing calculator. Assess communication skills as students write about graphing without a graphing tool to aid them. Activity 3.9: Graphing Non-Factorable Quadratic Equations in ax2 + bx + c form using the x-intercepts Time: 75 minutes Description To graph quadratic equations of the form y = ax2 + bx + c that cannot be factored, students will express the quadratic equation in the form y = ax(x - s) + t to locate two points and deduce the vertex. Strands and Expectations Ontario Catholic School Graduate Expectations The graduate is: - a reflective and creative thinker who evaluates situations and solves problems; - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills. Strand(s): Quadratic Functions Overall Expectations QFV.02 - determine through investigation, the basic properties of quadratic functions; QFV.04 - solve problems involving quadratic functions. Specific Expectations QF2.04 - sketch, by hand, the graph of a quadratic function whose equation is given in the form y = ax2 + bx + c, using a suitable method (e.g., locate the x-intercepts if the equation is factorable; express in the form y = ax(x - s) + t to locate two points and deduce the vertex); QF4.01 - determine the zeros and the maximum and minimum value of a quadratic function, using algebraic techniques. Planning notes Graphing calculators will be needed. The use of the calculator overhead display unit will enhance the visual part of the lesson and aid in the discussion about determining coordinates of the vertex. Prior Knowledge Required Common Factoring, finding the zeros of a quadratic using the graphing calculator. Teaching/Learning Strategies Teacher Facilitation: Begin with a class discussion on graphing the quadratic y = 3x2 - 6x + 2. Students will notice that the equation cannot be factored, and thus cannot be graphed using a method they already know. Lead the class through an analysis of related quadratics graphed using the graphing calculator and the overhead display. Unit 3 - Page 34 Principles of Mathematics - Academic Student Activity Using the graphing calculator, graph y = 3x2 - 6x using a graphing window with the dimensions -2 x 3 and -3 y 5. Examine the graph to determine the x-intercepts, and the coordinates of the vertex. Graph y = 3x2 - 6x + 2 on the same axis. Examine the graph, noting the coordinates of the vertex and the yintercept. How are the graphs related? (The graph is congruent to the original graph but vertically translated up 2 units, the x-coordinates of the vertices are the same, the y-coordinate of the vertex shows a translation upwards by 2.) Graph on the same axis y = 3x2 - 6x + 4 and compare to the original graph y = 3x2 - 6x. (This graph is congruent to the original graph, but vertically translated up 4 units.) Teacher Facilitation: To determine an algebraic method to draw the graph, lead the students to determine the x-intercepts by common factoring the 3x2 - 6x terms, translating the intercepts up 2 units, determining the x-coordinate of the vertex, and substituting into the equation to determine the ycoordinate of the vertex. Student Activity Graph y = 2x2 + 4x - 5 by hand using an algebraic method. Check your graph on the graphing calculator. Teacher Facilitation: Prompt the students to determine the x-intercepts by common factoring the first two terms to place the quadratic into the form 0 = 2x(x + 2) - 5. Without the translation, the x-intercepts would occur a (0, 0) and (-2, 0) but after translation down 5 units, the intercepts will become the points (0, -5) and (-2, -5). The students should see that the vertex occurs midway between these points (i.e., where x = -1). Thus, the x-coordinate of the vertex will be -1. Substituting x = -1 into the equation y = 2x2 + 4x - 5, the y-coordinate is calculated as -7, giving the vertex point as (-1, -7). Student Activity Practise graphing the following quadratics without the aid of a graphing calculator. (These quadratics differ from the examples already completed) Graph y = -3x2 - 9x - 1 without using technology (Parabola opens down, vertex (-1, 2)). Graph y = 5x2 - 15x + 2 without using technology (vertex has fractional value for x and requires substitution of a fraction into the equation to find y). Homework: From the textbook, practise graphing quadratics that cannot be factored as in the examples studied today. Assessment/Evaluation Techniques Use paper and pencil tasks to assess knowledge and understanding of graphing. Unit 3 - Page 35 Principles of Mathematics - Academic Activity 3.10: Max/Min Problems Time: 150 minutes Description Students determine the maximum and minimum values for quadratic relationships from realistic situations. The first part of the activity involves using paper and scissors to design a box with maximum surface area for a side of the box. The second activity involves an algebraic examination of the maximum area of different fenced shapes. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: -a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; -a collaborative contributor who works effectively as an interdependent team member; -a collaborative contributor who respects the rights, responsibilities and contributions of self and others. Strand(s): Quadratic Functions Overall Expectations QFV.03 - determine, through investigation, the basic properties of quadratic functions; QFV.04 - solve problems involving quadratic functions. Specific Expectations QF3.01 - collect data that may be represented by quadratic functions, from secondary sources, or from experiments; QF3.04 - report the findings of an experiment in a clear and concise manner, using appropriate mathematical forms (e.g., written explanations, tables, graphs, formulas, calculations), and justify the conclusions reached; QF4.01 - determine the zeros and the maximum or minimum value of a quadratic function, using algebraic techniques; QF4.02 - determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software; QF4.03 - solve problems related to an application, given the graph or the formula of a quadratic function. Planning Notes Students work in groups on the activities. Have grid paper available for students to cut out their model boxes in a variety of shapes. After completing the first activity, teachers may wish to practise more sample problems using paper and pencil examples from the text. The second part of the activity could be completed the next day, as another separate activity, or as an assessment activity. Prior Knowledge Required surface area formula, graphing quadratic functions using x-intercepts, quadratic regression on the graphing calculator Unit 3 - Page 36 Principles of Mathematics - Academic Teaching/Learning Strategies Teacher Facilitation: Pose this scenario: The students at Holy Name High School are raising money to help feed the poor in Third World countries. They want to design a coin collection box to place beside the cash registers at local businesses. They wish to design a box that will give them the largest possible surface area on the side, so that they can print in large letters STOP HUNGER. Squares of cardboard, 12 cm per side, have been donated to the school for the project. (For simplicity, assume the box is open at the top, and two identical open shapes fit together into each other to make the base and top of the box.) Student Activity Use paper and scissors to cut out boxes of various sizes. Determine the dimensions that will give the greatest surface area on the side. Is this a practical size for a collection box? Teacher Facilitation: Circulate around the room assisting students as they cut out various sizes of squares from the corner of the paper and fold into a box shape. Prompt them to record their data on a table, using the side of the cutout square as the independent variable and the area of the box side as the dependent variable. After the students determine the measures for maximum surface area, conduct a full class discussion. Plot the points from their tables using the LIST feature of the graphing calculator and use quadratic regression to determine the equation of the relationship. Graph the resulting equation and discuss (i.e., maximum because the parabola opens down). Lead students to see the need to find an algebraic method to solve the problem instead of a paper-cutting method. Model the problem algebraically: Surface Area of Side = x(12 - 2x) SA = 2x(6 - x) Graph intercepts at 0 and 6, and determine the vertex as (3, 18) From the graph, discuss maximum surface area is 18 when the side length, x, is 3. Part B: (This activity may be completed now, as a separate activity the next day, or as an assessment activity after students have practised a variety of max/min problems.) Teacher Facilitation: Present the following scenario: The Jackson family wants to build a fenced play area for their toddler. They purchased 24 m of wire fencing, but can’t decide where to place the play area so the child has the largest possible play area enclosed by the 24 m of fencing. They have 4 possible locations: 1) a rectangular fenced area in the middle of the yard, 2) against one wall of the house, 3) in the corner of the yard against the wooden fence, and 4) around the corner of the house. Unit 3 - Page 37 Principles of Mathematics - Academic Student Activity Determine which of the play areas will have the largest area. Prepare a full written report for the Jackson family, supporting your decision mathematically. Assessment/Evaluation Techniques Part B could be used as an individual assessment activity. The written report could be assessed by adapting the rubric for written reports found in Appendix D. Assess knowledge and understanding of max/min applications using paper and pencil tasks. Follow-up Skills: 150 minutes Solve problems involving the maximum and minimum values of quadratics. Activity 3.11: A Square Deal! Time: 75 minutes Description Students will use the algebraic method of completing the square to express the equation of a quadratic function in the form y = a(x - h)2 + k given the form y = ax2 + bx + c and sketch the quadratic function by hand. Algebra tiles will be used to concretely illustrate the concept of completing the square. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands and uses written materials effectively; - a self-directed, life-long learner who demonstrates flexibility and adaptability; - a collaborative contributor who works effectively as an interdependent team member; - a self-directed, life-long learner who applies effective communication, decision-making, problemsolving, time and resource management skills. Strand(s): Quadratic Functions Overall Expectations QFV.01 - solve quadratic equations; QFV.02 - determine through investigation, the relationships between the graphs and the equations of quadratic functions; QFV.04 - solve problems involving quadratic functions. Specific Expectations QF1.01 - expand and simplify 2nd degree polynomial expressions; QF2.03 - express the equation of a quadratic function in the form y = a(x - h)2 + k given it in the form y = ax2 + bx + c, using the algebraic method of completing the square in situations involving no fractions; QF4.01 - determine the zeros and the maximum or minimum value of a quadratic function using algebraic techniques; QF4.03 - solve problems related to an application, given the graph or the formula of a quadratic function. Planning Notes availability of graphing calculators or graphing software class set of algebra tiles and overhead set of algebra tiles Unit 3 - Page 38 Principles of Mathematics - Academic Prior Knowledge Required expanding polynomials, graphing quadratics using the form y = a(x - h)2 + k, area formula for rectangles, use of graphing calculators, use of algebra tiles an asset Teaching/Learning Strategies Teacher Facilitation: The following lesson consists of three distinct activities. In Activity A the students discover that the same parabola can have different forms for its equation. Activity B is a student activity that uses the algebra tiles to demonstrate the technique of completing the square. Activity C is a teacher-directed lesson on using the method of completing the square to convert y = ax2 + bx + c to the form y = a(x - h)2 + k. Student Activity Part A: Quadratic Twins Students may work individually or in pairs on this activity. Use a graphing calculator to complete the following: 1. Enter y = (x - 1)2 + 3 as equation Y1 = and graph. 2. Enter y = x2 - 2x + 4 as equation Y2 = and graph. (Note: Do not clear the first equation.) 3. Why is there still only one graph on the screen? 4. Support or refute your answer to question 3 by expanding and simplifying the equation in step 1. 5. Repeat the above steps using y = 2(x + 3)2 - 1 as Y1= and y = 2x2 + 12x + 17 as Y2 =. Teacher Facilitation: As students complete this short investigation, circulate and aid students who are experiencing difficulties with graphing calculators or with simplifying polynomial expressions. Upon completion of the investigation, lead a short discussion to summarize the relationship between the two forms of a quadratic relation emphasizing that if y = a(x - h)2 + k can be expanded to y = ax2 + bx + c then there must also be a method to convert y = ax2 + bx + c back to the form y = a(x - h)2 + k. Tell students that this algebraic method required is called “Completing the Square”. Now introduce the algebra tiles to illustrate the concept of completing the square by having students complete the next activity. Students should be familiar with algebra tiles if they used them previously for factoring trinomials in the Follow-up Skills to Activity 3.7. However, if this is the first time students have used algebra tiles, then a short explanation of the tiles and what they represent should be addressed before students complete the following activity. Part B: Completing the Square Using Algebra Tiles Students work individually or in pairs with a set of Algebra Tiles 1. Use algebra tiles and create the following rectangles, one at a time, having the areas of: a) 2x2 + 4x b) 3x + 6 c) x2 + 4x + 4 Make sure to draw a sketch of each rectangle. Determine the length and width of each rectangle. 2. What is unique about rectangle c? 3. Determine the tiles needed to create the polynomial x2 + 6x. Take these tiles and create a square by adding more tiles. Which tiles must you select and how many must be added to x2 + 6x to create an exact square? Draw a diagram and determine the length and width of the square created. Create an equation for the square’s total area. 4. Repeat step 3 for x2 + 2x and x2 + 8x. 5. This technique is known as “Completing the Square”. How does the number of unit tiles added to the given tiles compare to the number of x tiles present? 6. Determine a general rule to calculate the number of unit tiles needed to make x2 + bx a square. 7. Complete the square and factor (i.e., state the length and width) without using algebra tiles for: a) x2 + 10x b) a2 + 24a c) m2 + 16m d) c2 + 72c Unit 3 - Page 39 Principles of Mathematics - Academic Teacher Faciliation: There are two possible ways students could arrange tiles for step 1 part c as shown. Arrangement One: Arrangement Two: Insist that they arrange tiles like arrangement two. This will emphasize the idea that the x tiles must be divded into two equal groups (i.e., take half of the coefficent of the x term) before adding enough unit tiles to complete the square. This way the students should see the unit tiles added also form a square. While students are working on the activity, observe students and prompt when necessary. Conference with students to determine if the method of completing the square has been developed and understood. Conclude the activity and ensure that you draw out the rule for completing the square on x2 + bx as: Take half the value of b (i.e., the coefficient of x), square this value and add to the binomial to create a perfect square trinomial. This may need to be done after question 6 and before question 7 of the activity if several students are not able to come up with the rule of completing the square themselves. Discuss how to quickly factor this perfect square trinomial. Now lead through various examples to extend completing the square where b is a negative value and for binomials of the form ax2 + bx that require common factoring before the square can be completed. Part C: Teacher-directed Activity: Algebraic Method of Completing the Square Using a variety of examples show the algebraic method of completing the square on y = ax2 + bx + c to convert to the form y = a(x - h)2 + k. Use the two examples from the first activity ‘Quadratic Twins” and show how the general form can be converted to graphing form. Use only examples that involve no fractions (e.g., y = x2 + 10x - 3, y = -3x2 + 12x, y = x2 - 16x + 2, y = 5x2 - 40x + 13, etc.). Remember to emphasize the fact that once the constant value to complete the square has been added to the equation then it must also be subtracted to maintain equality for the equation. Students are now working with equations not just expressions as in Activity B. Once the form has been converted students may use the calculator and graph both forms of the quadratic on the same set of axes to verify that these equations produce the same graph. Reinforce reading the maximum or minimum value from the form y = a(x - h)2 + k. Review the methods used to graph quadratic functions and discuss the advantages and disadvantages of each. For instance consider y = x2 - 6x + 8 or y = 3x2 - 6x and sketch the graph by a) finding the xintercepts by factoring then find vertex or b) completing the square. It is hoped that students will use completing the square as a last resort. Alternate Activity: The investigation could be performed as a demonstration using overhead algebra tiles if there are insufficient tile sets for the entire class. However algebra tiles can easily be made from bristol board if funds do not allow for the purchase of actual manufactured sets. Unit 3 - Page 40 Principles of Mathematics - Academic Assessment/Evaluation Techniques A paper and pencil quiz could assess the student’s knowledge and understanding of converting a quadratic equation to standard vertex form by completing the square. Communication and Understanding can be assessed by conferencing with students during the completion of the investigations. Initiative, teamwork, and independent work could be assessed using the rubrics found in the Appendices. Follow-up Skills: 150 minutes QF4.03 - solve problems related to an application, given the graph or the formula of a quadratic function Revisit max/min problems and solve by completing the square. This way the problem can be solved by reading the max/min value from the equation after completing the square rather than sketching the quadratic first to locate the vertex. Extensions: The method of completing the square may also be used to solve quadratic equations. This topic is not an expectation of the course but could be used as an alternative method to factoring if desired. This technique will be used in the next activity to develop the quadratic formula. Resources Scully, J., B. Scully, and J. LeSage. Active Learning Series “Alge-Tiles”. Exclusive Educational Products. Activity 3.12: Root of the Problem Time: 75 minutes Description Students will develop the quadratic formula to solve quadratic equations specifically for the cases where the quadratic equation can not be factored. Students will do a short activity which will lead to a necessity to develop a method for solving non-factorable quadratic equations. The quadratic formula will be developed solving the general equation y = ax2 + bx + c using the method of completing the square. Strand(s) and Expectations Ontario Catholic Graduate Expectations The graduate is expected to be: -an effective communicator who reads, understands and uses written materials effectively; -a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; -a self-directed, life-long learner who demonstrates flexibility and adaptability. Strand(s): Quadratic Functions Overall Expectations QFV.01 - solve quadratic equations. Specific Expectations QF1.04 - solve quadratic equations, using the quadratic formula; QF1.05 - interpret real and non-real roots of quadratic equations geometrically as the x-intercepts of the graph of a quadratic function. Unit 3 - Page 41 Principles of Mathematics - Academic Planning Notes Have graphing calculators available or graphing software Use overhead projection of calculator especially if a class set of calculators is not available Prior Knowledge Required solving quadratic equations by factoring, sketching quadratic functions by hand, completing the square, distinguish between real and non-real roots graphically Teaching/Learning Strategies Student Activity: Root of the Problem 1. For each quadratic equation determine the x-intercepts algebraically and sketch the quadratic. Verify the graph using the graphing calculator. a) y = x2 - 4x - 21 b) y = 3x2 - 12x c) y = (x + 4)2 2 2. Graph y = x - 5x + 3 using the graphing calculator. How many x-intercepts does this graph have? 3. Determine the roots (x-intercepts) of y = x2 - 5x + 3 algebraically. What problem do you encounter? 4. Find the x-intercepts of y = x2 - 5x + 3 using the CALCULATE ZERO function on the calculator. Teacher Facilitation: The first three equations of the activity should be solved quickly since this is review from previous activities. The instructions for the CALCULATE ZERO function can be found in Activity 3.7 if a refresher is needed. Circulate and observe students’ ability to solve and graph quadratic equations. Together with students conclude that even though y = x2 - 5x + 3 has two roots as shown by the graph on the calculator and the calculator can calculate the value of the roots, algebraic calculation of the roots cannot by performed since the trinomial cannot be factored. This leads to the teacher-directed development of the quadratic formula. Develop the quadratic formula by simultaneously solving the equation 2x2 - 14x + 9 = 0 and the equation 0 = ax2 + bx + c beside each other to allow for comparison of the algebraic method of completing the square on a numerical equation versus an abstract general equation. Note: This development must include the use of fractions when completing the square. Students have not been exposed to this, therefore it is crucial that the teacher carefully explains each step clearly. Also if students have not solved equations by completing the square (an extension of the previous activity) the method will be new. The intent of this development is to show how the quadratic formula was created. Students will not be expected to reproduce this development. This development could be pre-written on an overhead acetate showing both solutions side by side and the teacher could reveal the solutions line by line to compare the algebra involved. This could be used as a demonstration and only the final formula could be recorded in student notebooks along with other examples using the formula to solve for roots. Development of the Quadratic Formula Solve for the x-intercepts by the method of completing the square for each quadratic function. 2x2 + 14x + 9 = y ax2 + bx + c = y let y = 0 let y = 0 2 2x + 14x + 9 = 0 ax2 + bx + c = 0 2(x2 + 7x) + 9 = 0 (x2 + 7x) = x2 + 7x + ( (x + 9 2 7 2 9 7 ) = - + ( )2 2 2 2 7 2 49 9 ) = 2 4 2 Unit 3 - Page 42 b x) + c = 0 a b c (x2 + x) = a a b c b b x2 + x + 2 = + 2 a a 2a 2a a(x2 + (x + 2 c b 2 ) = b2 a 2a 4a Principles of Mathematics - Academic 7 2 49 18 ) = 2 4 (x + x+ 7 31 = 2 4 x=x= 7 2 31 2 7 31 2 (x + b 2 b 2 4ac ) = 2a 4a 2 x+ 2 4ac b = b 2 2a 4a x=x= b 2a b2 4ac 2a b b2 4ac 2a Once the formula has been developed, return to the Student Activity: Root of the Problem and use the formula to solve exactly for the roots of the last example y = x2 - 5x + 3. Show how to convert the radical form of the roots to the decimal approximations and have students compare these answers to the roots that the calculator gave for this quadratic. Provide a series of examples of quadratic equations to solve using the formula. Practise reading the values of a, b, and c and how to appropriately substitute these values into the quadratic formula. Be sure to include examples where the quadratic has only one root (i.e., the value b2 - 4ac under the root equals zero) and has no real roots (i.e., the value b2 - 4ac equals a negative integer). Students should graph the quadratics to verify how the solutions from the quadratic formula relate to the nature of the graph. Examples should also include equations that can be factored. Suggest that the formula be reserved for trinomial equations that cannot be factored (i.e., try to solve by factoring first before resorting to the quadratic formula). Extension: The discussion of the number and nature of roots could be introduced by viewing the value of the discriminant (expression of b2 - 4ac under the root sign) rather than solving completely for the roots and then making a decision on the number of real roots a quadratic will have. This topic is not an expectation of this course but could be used for enrichment students. The graphing calculator can also be used to solve quadratic equations using the quadratic formula. Please refer to the instruction manual that accompanies your specific calculator for these instructions if so desired. Assessment/Evaluation Techniques A paper and pencil quiz can assess the knowledge and understanding of using the quadratic formula to solve equations. Assess Problem Solving and Inquiry Skills by monitoring communication skills of students as they interact with each other to complete the investigation and how they effectively make use of the graphing calculator. Follow-up Skills: 60 minutes Practise solving equations using the quadratic formula. Interpret the number of roots and whether the roots will be real or non-real from the solutions obtained algebraically. Unit 3 - Page 43 Principles of Mathematics - Academic Activity 3.13: Summative Assessment Activities Time: 150 minutes Description This unit assessment includes a paper and pencil test designed by the classroom teacher as well as a graphical investigation based on material learned and extended from the quadratic unit. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a collaborative contributor who achieves excellence, originality and integrity in one’s own work and supports those qualities in the work of others; - a responsible citizen who contributes to the common good. Strand(s): Quadratic Functions Overall Expectations: All Overall Expectations from the Quadratic Strand. Specific Expectations: All Specific Expectations from the Quadratic Strand. Planning Notes Students will need graphing calculators or graphing software to complete part of the activity. Teaching/Learning Strategy Student Activity Students individually complete the following investigation using a calculator or graphing software. Determine the Effect of Varying a, b, or c in the quadratic y = ax2 + bx + c 1. Complete the following investigation to determine the effect of varying b in y = ax2 + bx + c, when the values of a and c remain fixed such that a = 1 and c = 2: Y = x2 + 2 Y = x2 - 2x + 2 2 Y = x + 2x + 2 Y = x2 - 4x + 2 2 Y = x + 4x + 2 Y = x2 - 6x + 2 Y = x2 + 6x + 2 Determine, from examining each graph, the coordinates of each vertex and the y-intercept of each graph. Plot all 7 vertices. What type of relationship do the 7 vertices form? (parabola) Support your claim by using a table of values. Determine an equation of the relationship using the table. Verify the accuracy of the equation using the regression capabilities of the calculator. Make conclusions about the quadratic y = x2 + bx + 2 as b varies. (As b varies, a variety of parabolas are generated. The vertices of all the parabolas form another distinct parabola with the equation y = - x2+ 2) Make a general conclusion about the quadratic y = x2 + bx + c as b varies. (As b varies, a variety of parabolas are generated. The vertices of all the parabolas form another distinct parabola with the equation y = - x2+ c) Make a hypothesis about the graphs of the quadratic y = ax2 + bx + c as b varies while a and c are fixed, but a 1. Design and carry out an investigation to verify or refute your hypothesis. a) On the same axes, graph b) c) d) e) f) g) h) Unit 3 - Page 44 Principles of Mathematics - Academic 2. Complete the following investigation to determine the effect of varying c in y = ax2 + bx + c, when a = 1 and b remains fixed: a) On the same axes graph Y = x2 + 4x + 2 Y = x2 + 4x + 1 2 Y = x + 4x + 5 Y = x2 + 4x - 1 2 Y = x + 4x - 2 Y = x2 + 4x - 5 b) Determine vertices, y intercepts and axes of symmetry of each parabola. c) Make conclusions about the effect of varying c in y = x2 + 4x + c. d) Describe how the values of b and c are related to points on the parabola. (b is -½ of the xcoordinate of the vertex, c is the y-intercept) e) Make a hypothesis about the graphs of the quadratic y = ax2 + bx + c as c varies while a and b are fixed, but a 1. Design and carry out an investigation to verify or refute your hypothesis. Assessment/Evaluation Techniques Assess the student activity using the Rubric for Assessing Written Reports with the following criteria for the categories of the Achievement Chart: Assess Knowledge using the expectation QF3.02 - fit the equation of a quadratic function to a scatterplot, using an informal process, and compare the results with the equation of a curve of best fit produced by using graphing calculators or software. Assess Problem Solving using the expectation QF2.01 - identify the effect of simple transformations on the graph and the equation of y = x2, using graphing calculators or graphing software. Assess Communication Skills using the expectation QF3.04 - report the findings of an experiment in a clear and concise manner, using appropriate mathematical forms, and justify the conclusions reached. Unit 3 - Page 45 Principles of Mathematics - Academic Unit 4: Similarity and Applied Trigonometry Time: 24 hours Unit Description In this unit, students will investigate the properties of similar and congruent triangles and their use in modelling realistic situations. Students will develop and investigate the primary trigonometric ratios using technology. Right-angled triangles will be used to measure the heights of inaccessible objects around the school. Students will apply trigonometric ratios, the sine law, and the cosine law to solve realistic problems in acute-angled triangles. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be an effective communicator who: - reads, understands and uses written materials effectively; presents information and ideas clearly and honestly and with sensitivity to others; - 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. The graduate is expected to be a reflective and creative thinker who: - thinks reflectively and creatively to evaluate situations and solve problems; -evaluates situations and solve problems; -adopts a holistic approach to life by integrating learning from various subject areas and experience. The graduate is expected to be a collaborative contributor who: - works effectively as an interdependent team member; - exercises Christian leadership in the achievement of individual and group goals. The graduate is expected to be a self-directed, responsible, life long learner who: -applies effective communication, decision-making, problem-solving, time and resource management skills; - demonstrates flexibility and adaptability; - sets appropriate goals and priorities in school, work and personal life; - examines and reflects on one’s personal values, abilities and aspirations influencing life’s choices and opportunities. The graduate is expected to be a responsible citizen who: -respects the environment and uses resources wisely. Strand(s): Trigonometry Overall Expectations: All those from the Trigonometry Strand Specific Expectations: All those from the Trigonometry Strand Activity Titles (Time + Sequence) The following table provides a suggested sequence for teaching Unit 4. After (or during) each activity, the skills to be developed are stated, as well as suggested timing to work on skill proficiency. This unit could be done before or after any of the other units, and need not be left until the end of the course. Because of the outdoor field trip component in this unit, teachers may choose to teach this unit when the weather best allows for outdoor activities. The activities in the unit are designed to help students visualize and analyse the use of trigonometry in the world around them, as well as promote a respect for God’s creation and an understanding of the need to use resources wisely. The career investigation provides an opportunity for students to examine and Unit 4 - Page 1 Principles of Mathematics - Academic reflect on their personal values, abilities and aspirations as they influence life’s choices and opportunities. Activity 4.1 75 minutes It’s Similarity, My Dear Watson! Use The Geometer’s Sketchpad™ to investigate and summarize the properties of similar triangles Follow-up Skills Setting up proportions to solve for missing lengths, determine 150 minutes areas of similar triangles, solve problems with realistic situations using similar triangles, distinguish between similarity and congruency Activity 4.2 75 minutes Schoolyard Field Trip Students determine the height of inaccessible objects using similar triangles and using congruent triangles. Activity 4.3 90 minutes Sine Field Group activity comparing the lengths of sides of right triangles and summarizing the results to develop the definitions of sine, cosine, and tangent. Follow-up Skills Practise solving right triangles by finding angles and sides, 225 minutes applications of right triangles (including use of Pythagorean theorem), angles of elevation and depression, reciprocal trigonometric ratios Activity 4.4 75 minutes Let’s Go to the Movies! Use tangent ratios in overlapping triangles to determine the best distance from a movie screen to allow the largest viewing area (maximize the angle). Activity 4.5 75 minutes Sines, Sines, Everywhere Sines! Use The Geometer’s Sketchpad™ or paper and pencil calculations to determine triangle inequalities and the Sine Law Follow-up Skills Practise using sine law to solve problems, revisit overlapping 300 minutes triangle solution to the Movies problem, introduce cosine law when sine law cannot be applied, practise using cosine law Activity 4.6 75 minutes Touring With Trigonometry Use trigonometry to solve practical problems involving lengths and angles within the school environment. Follow-up Skills Practise solving triangles that use combinations of all of the 75 minutes trigonometric methods studied thus far. Activity 4.7 150 minutes Trigs of the Trade Students research a career or application that uses trigonometry. Research is presented at a Math Fair. 75 minutes Assessment: Paper and Pencil Test Prior Knowledge Required Ratio and rate skills, equation solving skills, properties of angles in a triangle, properties of angles in parallel lines, use of The Geometer’s Sketchpad™ Unit 4 - Page 2 Principles of Mathematics - Academic Unit Planning Notes Check the availability of the computer lab for Activities 4.1 and 4.5. Students will work in pairs on these activities using The Geometer’s Sketchpad™. After each investigation, the teacher should ensure that the mathematics to be developed during the activity has been drawn out. Plan enough time at the end of each investigation to bring closure to the activity, and perhaps write a concluding note or journal entry. Teaching/Learning Strategies This unit provides opportunities for a balance of whole-class, small group, and individual instruction through student-centred and teacher-directed activities. The unit uses rich contextual problems which engage students and provide them with opportunities to demonstrate achievement of the Trigonometry Strand Expectations. The use of technological tools and software (eg., dynamic geometry software, Internet, spreadsheets) are used to facilitate the exploration and understanding of mathematical concepts. The students are encouraged to practise and extend their skills and knowledge outside the classroom in the form of two field trips and external research completed on a topic related to trigonometry. Assessment and Evaluation A balance of assessment tools and strategies is recommended. The suggested assessment at the end of this unit is an individual paper and pencil skills test. A variety of other assessment techniques are implemented throughout the unit to assess learning skills and the four categories of the Achievement Chart. Rubrics are found in the Appendices and should be used when the students perform the openended activities. (In the Course Overview, a variety of assessment tools and strategies are listed.) Resources Web Sites www.coolmath.com/careers.htm http://forum.swarthamore.edu Books Exploring Trigonometry With Geometer’s Sketchpad Unit 4 - Page 3 Principles of Mathematics - Academic Activity 4.1: It’s Similarity, My Dear Watson! Time: 75 minutes Description Students use The Geometer’s Sketchpad™ to discover the properties of similar triangles. Using a pair of similar triangles, students will determine the degree measurements of all angles, the lengths of sides and the areas of each triangle. Comparisons are then made among the corresponding angles, corresponding sides lengths, and areas of the triangles in order to determine the relationships that exist for angles and sides of similar triangles. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands, and uses written materials effectively; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a self-directed, responsible, life-long learner who demonstrates flexibility and adaptability. Strand(s): Trigonometry Overall Expectations TRV.01 - develop the primary trigonometric ratios, using the properties of similar triangles. Specific Expectations TR1.01D - determine the properties of similar triangles (e.g., the correspondence and the equality of angles, the ratio of corresponding sides, the ratio of areas) through investigation, using dynamic geometry software Planning Notes Reserve computer lab time to use The Geometer’s Sketchpad™. Modification of this activity is included where availability of technology is limited. Prior Knowledge Required ratio; use of The Geometer’s Sketchpad™ Teaching/Learning Strategies Teacher Facilitation: Pose the following scenario to the class: On a bright sunny day, you and a group of friends are flying a hot-air balloon. Your friends are in the balloon and you are riding in a truck with Watson, the guide, tracking the flight path of the balloon from ground level. It is the job of the guide to drive ahead of the balloon to check for any obstructions that may be in the balloon’s flight path. The balloon is flying at a constant height. Watson suddenly notices that the town’s oldest oak tree, a landmark for centuries, is in the path of the approaching balloon. He needs to determine the height of the tree quickly so he can radio the balloon’s navigator to change altitude if necessary. All that is found in the truck is a tape measure and a broom. How can these items be used to determine the height of the tree and prevent a possible collision? Unit 4 - Page 4 Principles of Mathematics - Academic Student Activity: Investigation of Similar Triangles Using The Geometer’s Sketchpad™ Teacher Instruction The Geometer’s Sketchpad™ Instruction Create a graph grid. Graph, Show grid Plot the following three points to create the Graph, Plot Points and enter coordinates (2, 4), vertices of a triangle. (-4, 2), (6, -2) using the Tab key, OK. Join the points to form the sides of a triangle. Select points in pairs. Construct, Segment Repeat to construct a second triangle. Graph, Plot Points (-0.5, -1), (1, -0.5), (-1.5, 0.5). and enter coordinates using the Tab key, OK. Visually compare the two triangles. What relationships do you notice? Measure the lengths of all sides. Select a side and choose: Measure, Length. Determine the degree measurements of all angles Hold the SHIFT key as you select the three in each triangle. vertices in the proper order then choose: Measure, Angle. Calculate the area of each triangle. Hold the SHIFT key as you select the three vertices of the triangle. Choose: Construct, Polygon Interior, Measure, Area. What do you notice about the angle measurements in each triangle? Compare the ratios of the lengths of the sides of Select two sides and choose Measure, Ratio the triangles. Compare the large triangle’s sides to the small triangle sides in sets of two as follows: Longest side in large triangle to longest side in small triangle; shortest sides together and the remaining two sides together. What do you notice about the ratio of each pair of sides? Compare the area of the large triangle to the area of the small triangle by creating a ratio. How does this ratio compare to the ratio of side lengths? Repeat the above procedure for another two triangles formed by the following vertices: (-6, -3), (-3, 0), (3, 3) and (3, -3), (4, -2), (6, -1). The pairs of triangles you have examined are called SIMILAR TRIANGLES. Write a statement about angles of similar triangles. Write a statement about corresponding sides of similar triangles. Examine two triangles where one triangle is a Construct a triangle using any three points. dilatation of the other. Two overlapping triangles Select one of the points and choose Transform, will appear on the screen, one triangle being a Mark Center to set the center of the dilatation. dilatation of the other. Select the triangle, choose Transform, Dilate, Scale Factor 2. Are the two overlapping triangles similar? Verify by determining the ratio of corresponding sides and measuring corresponding angles. Examine two triangles where one triangle is a Construct a triangle using any three points. rotation of the other. Two triangles will appear on Select one of the points and choose Transform, the screen, one triangle being a rotation of the Mark Center to set the center of the rotation. other. Select the triangle, choose Transform, Rotate, and choose any angle size for the rotation. Verify by determining the ratio of corresponding sides and measuring corresponding angles. Leave the measures on the computer screen. Drag one of the vertices of the triangle. What happens to the ratio of the sides when the angle is altered? Unit 4 - Page 5 Principles of Mathematics - Academic Teacher Facilitation: While students are working on the investigation, the teacher should circulate about the room to aid and prompt students who are experiencing difficulties. Once students have finished the investigation, the teacher should lead the entire class in summarizing the properties of similar triangles emphasizing: corresponding angles are equal corresponding sides are proportional the ratio of the areas is the square of the ratio of corresponding sides Demonstrate the proper notation for writing the similarity statement so that the proportional statement follows from it. Provide sufficient practise setting up similarity and proportional statements. Return to the initial problem about the hot-air balloon and brainstorm with the students how a tape measure, a broom, and the properties of similar triangles could be used to determine the height of the old oak tree. (Students may need more explanation to understand that overlapping right triangles are similar when one side of the triangle represents the sun’s rays.) This could also be assigned for homework. Alternate Activity Where Technology is Limited If dynamic geometry software is not available the activity can be performed using graph paper, rulers, and protractors. Students can plot the points given and calculate the lengths of line segments using the distance formula between two points. Angles can be measured using a protractor. Areas may be calculated using the area formula for triangles. To facilitate area calculations, select coordinates that would produce right scalene similar triangles so that the base and height are easily identified. Assessment/Evaluation Techniques The teacher can assess knowledge and understanding by conferencing with students about their findings as the students are working through the investigation. Communication can be assessed by having students prepare a solution to the hot-air balloon scenario through journal writing or by presenting a report of the solution to the class. Assess the written report using the rubric found in Appendix D, and use the rubric in Appendix C to assess the verbal report. Learning skills, specifically teamwork, independence, and initiative, can be assessed using the rubric in Appendix A. Follow-up Skills: 150 minutes TR1.02 - describe and compare the concepts of similarity and congruency TR1.03 - solve problems involving similar triangles in realistic situations (e.g., problems involving shadows, reflections, surveying) Resources Exploring Geometry with the Geometer’s Sketchpad. Key Curriculum Press. Activity - “Similar Triangles - AA Similarity” Unit 4 - Page 6 Principles of Mathematics - Academic Activity 4.2: Schoolyard Field Trip Time: 75 minutes Description Students apply their knowledge of similar and congruent triangles to determine the height of an inaccessible object in the immediate school environment. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a collaborative contributor who works effectively as an interdependent team member; - a responsible citizen who respects the environment and uses resources wisely; - a collaborative contributor who exercises Christian leadership in the achievement of individual and group goals. Strand(s): Trigonometry Overall Expectations TRV.01 - develop the primary trigonometric ratios, using the properties of similiar triangles. Specific Expectations TR1.03 - solve problems involving similar triangles in realistic situations; TR1.02 - describe and compare the concepts of congruence and similarity. Planning Notes The teacher locates appropriate objects outside the school building for which students will calculate height (e.g., church or school steeple or crucifix, football standards, tall tree, light post, etc.). A sunny day is required for this activity, as the students must measure lengths of shadows to calculate height using similar triangles. A supply of metre sticks, measuring tapes and/or trundle wheels is needed. Prior Learning Required properties of similar triangles, meaning of congruent triangles Teaching/Learning Strategies Teacher Facilitation: Some discussion of similar triangles and their application in calculating height of an object is needed. (Many sample questions can be found in textbooks.) The use of a stick (or pencil) as a measuring tool to calculate height using congruent triangles must be discussed and demonstrated. Student Activity Students work outside the school building taking measurements to determine the height of the inaccessible objects. The height will be calculated using both similar triangles and congruent triangles. Students prepare a report on the height, explaining and illustrating their methods, stating their solution, and analyzing any discrepancies between the height as calculated by the two different methods used. Unit 4 - Page 7 Principles of Mathematics - Academic How to Use Similar Triangles to Measure the Height of an Inaccessible Object (a tree): On a sunny day, measure the shadow of the tree along the ground. Place a pole near the tip of the tree’s shadow. Measure the height of the pole and the length of the pole’s shadow. Assuming that the sun’s rays are parallel, use similar triangle ratios to determine the height of the tree. On a cloudy day, use a mirror placed flat on the ground. Stand in a position where you can see the reflection of the top of the tree in the mirror. (Discuss angle of reflection and angle of incidence) How to Use Congruent Triangles to Measure the Height of an Inaccessible Object (a tree): Hold a stick in your line of vision of the tree, such that the top and bottom of the tree are lined up with the top and bottom of the stick. Without moving your location or altering your arm stretch, rotate the stick so that it is placed horizontally, with the left end of the stick visually lined up touching the base of the tree. Instruct your friend to begin walking away from the tree in a direction perpendicular to your line of sight of the tree. Tell him to stop walking when he visually reaches the tip of stick. With a tape measure, determine the distance that your friend is standing from the tree. This distance is equal to the height of the tree. Homework: Other applications of similar and congruent triangles as found in the textbook. Unit 4 - Page 8 Principles of Mathematics - Academic Assessment/Evaluation Techniques Assess Learning Skills such as teamwork and initiative by observing students as they are performing this activity. Assess Communication and Understanding by focussing on the student’s description of the method used, the appropriate use of terminology and diagrams, the clarity of the explanation, the accuracy of the solution and the analysis of any discrepancies. Activity 4.3: Sine Field Time: 90 minutes Description Students develop the primary trigonometric ratios: sine, cosine and tangent using right triangles. The activity involves measuring lengths of sides of triangles and measuring the acute angle formed between the hypotenuse and the adjacent side to the angle. There is a crucial need for exact measurements and the calculation of ratios. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others; - a collaborative contributor who works effectively as an interdependent team member; - a collaborative contributor who exercises Christian leadership in the achievement of individual and group goals; - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills. Strand(s): Trigonometry Overall Expectations TRV.01D - develop the primary trigonometric ratios, using the properties of similar triangles. Specific Expectations TR1.04D - define the formulas for the sine, the cosine, and the tangent of angles, using the ratios of sides in right triangles. Planning Notes Provide: chalkboard protractor for each group of four students; various sticks of different lengths with a string, at least as long as the stick, attached to one end; metre sticks where the string is attached at different positions (e.g., 45 cm, 60 cm, 75 cm, etc.) to simulate sticks of various lengths, if desired; tape measures or metre sticks (additional to ones with string attached); a tiled floor (tiled floor works best, however, any area with straight lines can be used, e.g., football field); summary data sheet for students and for group comparisons; scientific calculators. Unit 4 - Page 9 Principles of Mathematics - Academic Prior Knowledge Required properties of similar triangles; measurement skills with protractors, metre sticks, tape measures; Pythagorean theorem Teaching/Learning Strategies Teacher Facilitation: Prior to the activity the teacher will explain how to label the sides of a right triangle with respect to an acute angle as the opposite, adjacent, or hypotenuse side. Demonstrate the creation of the right triangle with student assistants. Each group of students is given a different hypotenuse length (stick with string). The students create a right triangle using a specific acute angle with the stick, the string and a floor line of a tiled floor. The lengths of the opposite and adjacent sides of the triangle are measured and are used along with the hypotenuse to create different ratios. The process is repeated for several acute angles. Student Activity Students will work in groups of four to design right triangles, measure and gather data and record results on a class summary data sheet. 1. Equipment: a stick with a string attached, two protractors, tape measure/metre stick, recording chart 2. Assign a duty to each group member as follows: a) angle measurer (positions protractor and one end of the stick) b) stick positioner (at opposite end of the stick from the angle measurer) c) string positioner (pull string straight from the stick so that it meets the base line at 90 degrees) d) distance measurer and data entry(measures the opposite and adjacent sides of the right triangle and enters data) 3. Lay the protractor flat along a floor line of a tiled floor. The floor line will represent the base of the triangle and the adjacent side to the acute angle. 4. Lay the stick flat and position it so that an acute angle of 30 degrees is formed between the floor line and the stick. The angle measurer and the stick positioner hold the stick at either end. The string positioner will pull the string straight down from the stick until it meets the base (floor line) at 90 degrees to form a right triangle. A protractor will be handy here as well. Unit 4 - Page 10 Principles of Mathematics - Academic 5. The distance measurer uses a tape measure or metre stick to accurately measure the opposite side (string length) and the adjacent side (baseline from angle to string) of the right triangle. The measurements are recorded in a table. Data Summary Chart Measurements Trigonometric Ratios acute angle length of opposite adjacent opposite adjacent opposite (degrees) hypotenuse side length side length hypotenuse hypotenuse adjacent 30 45 60 75 80 6. Repeat Steps 2-4 for other degree measurements of the acute angle. 7. After all measurements are taken complete the chart by calculating the ratios indicated to two decimal places. 8. Transfer your results for each angle on the summary sheets posted in the classroom. Teacher Facilitation: The teacher circulates among the groups to observe teamwork. Ensure that students are making accurate measurements and aid in further clarification when needed. Monitor the group work. Provide summary sheets for the sine, cosine and tangent ratios for each angle measured so that group results can be compared. Once students have completed the activity discuss the findings of the ratios of the 30-degree angle. The students should conclude that the ratios are the same regardless of the groups’ measurements. The same should be true for all the other angle measurements. It should be noted that all of the triangles that were created for a specific angle measurement were similar triangles. Since these ratios are always the same, then specific names have be assigned to these ratios. The teacher will introduce the names of the ratios as sine, cosine and tangent and define them accordingly. Students should now verify the sine, cosine, and tangent of the angles from their activity with a calculator to see how accurate their findings were. Be sure to discuss the importance of having the calculator in degree mode. Adequate practice should be given with a variety of right triangles to set up the sine, cosine, and tangent ratios of select acute angles. Emphasize the fact that the opposite and adjacent labels interchange depending on the acute angle used as a reference but the hypotenuse is always the longest side across from the right angle. Given a ratio, the student should be able to set up the sides of the right triangle using the trigonometric ratio and Pythagorean theorem. Assessment/Evaluation Techniques A paper and pencil test/quiz can assess the knowledge and understanding of the primary trigonometric ratios. Assess Learning Skills of teamwork and initiative through observation of group activity and use of a group work rubric. Assess Problem Solving and Inquiry Skills by monitoring communication skills of students as they interact within their group. Accommodations Since accurate measurements are essential to this activity, students experiencing difficulty with accuracy could use The Geometer’s Sketchpad™ to simulate the activity. (Instructions for using The Geometer’s Sketchpad™ for this activity can be found in the Applied 10 profile.) Unit 4 - Page 11 Principles of Mathematics - Academic Follow-up Skills: 225 minutes TR2.01 - determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios. TR2.02 - solve problems involving the measures of sides and angles in right triangles (e.g., surveying, navigation). Resources Exploring Trigonometry with the Geometer’s Sketchpad. Key Curriculum Press. Activity 4.4: Let’s Go to the Movies! Time: 75 minutes Description Students use mathematical modelling and the trigonometry of overlapping triangles to determine where they would sit in a movie theater to obtain the maximum viewing angle of the movie screen. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a self-directed, responsible, life long learner who demonstrates flexibility and adaptability; - a reflective and creative thinker who evaluates situations and solve problems; - a reflective and creative thinker who adopts a holistic approach to life by integrating learning from various subject areas and experience. Strand(s): Trigonometry Overall Expectations TRV.02 - solve trigonometric problems involving right triangles. Specific Expectations TR2.01 - determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios; TR2.02 - solve problems involving the measures of sides and angles in right triangles. Planning Notes The scenario posed could be adapted to your own school environment to model a situation that could be checked by measurement when the activity is completed. Tape measures and metre sticks are necessary. Students might work in pairs or individually. Since students will complete the problem based on their own eye levels, they should work with a person of the same height. Prior Learning Required using the tangent ratio in right-angled triangles angle of elevation using the trigonometric function keys on the scientific calculator Note: This activity should be completed before sine and cosine laws are introduced. Unit 4 - Page 12 Principles of Mathematics - Academic Teaching/Learning Strategies Teacher Facilitation: Present the following scenario: “If you arrived early enough to pick the best seat in the movie theatre, where would the seat be located?” Lead a discussion, ensuring that students determine that the best seat for viewing the screen would have the angle that provides the largest view of the screen (i.e., maximize the angle). Model the problem by placing a poster on the classroom wall, well above eye level. Students are to determine the best spot from which to view the poster, by determining the location that provides the largest viewing angle. It will not initially be obvious to students that the farther away they stand from the poster, the smaller the viewing angle will be. As they walk towards the poster, the viewing angle will increase in size, and then decrease to almost zero as they reach the wall. Choose one viewing position and examine the resulting triangle that can be drawn from that position. Brainstorm a method that could be used to determine the measure of the viewing angle. It is most likely that the teacher will need to introduce the overlapping triangle diagram when the students fail to see any right-angled triangles. Students will need to know the height of the top and bottom of the poster, as well as the height of their own eye level. Lead students to find the viewing angle, V, using these steps: (i) calculate the smaller angle, S, using the tangent ratio; (ii) calculate the larger angle, L, using the tangent ratio; (iii) subtract the smaller angle, S, from the larger angle, L. At their desks, students must calculate the viewing angle for a variety of distances, and should notice the angles increase and decrease in size as the distance varies. Student Activity Students make a hypothesis about the best viewing spot. Students then work individually or in pairs using their calculators to calculate the distance (correct to two decimal places) that will yield the largest possible viewing angle. Each student (or pair of students) must determine the angle for his/her own eye level. Students complete individual reports, drawing and labelling separate diagrams for each distance, writing a full solution for each diagram and organizing the data into a table to support their conclusion. Unit 4 - Page 13 Principles of Mathematics - Academic Teacher Facilitation: Circulate around the room as the students begin their calculations, prompting as necessary. Remind them to continue their calculations until they have determined the distance accurately to 2 decimal places. A table may be useful, as illustrated below. (Individual student measures will vary, depending on height and size of poster, as well as height of student.) Distance from 1m 2 m 3 m 4 m Best viewing angle 2.5 m 2.7 m 2.6 m poster appears between 2 m Viewing Angle 36° 43° 38° 34° and 3 m. Next, try 40° 41.9° 40.7° decimal measures. Extension 1. Would a shorter person stand closer or farther away from the poster? 2. Determine other situations where the maximum angle would be important to know. (Art galleries place barriers so that people can view, but not touch, the paintings; Jumbotron at Sky Dome) 3. Determine the spot on a basketball court that will yield the largest angle within which to throw a foul shot. How is this spot related to the free throw line? 4. Media Studies students may choose to illustrate the difference in angle views by preparing a video. 5. How would you allow for the sloped seating designs used in many theaters? 6. How does the width of a movie theatre screen affect the viewing angle? (i.e., if the screen is very wide, is there ever a situation where you would need to sit farther back to see the entire screen?) Homework: Textbook questions that feature overlapping triangles in a variety of situations. Include questions that provide angles of elevation to the right side and left side of an object. Assessment/Evaluation Techniques Assess Knowledge and Understanding in the written report by examining the accuracy of the calculations, any assumptions made during the investigation, and the correctness of the final solution. Assess Problems Solving and Application by observing students as they plan, discuss, gather data and make conclusions from the data. Assess Communication by examining the mathematical form used to express the trigonometric calculations, and to display the data coherently and logically. Assess the quality of the written report for effective mathematical terminology and well explained conclusions. Assess Learning Skills such as teamwork and initiative using a rubric. Activity 4.5: Sines, Sines, Everywhere Sines! Time: 75 minutes Description Students will determine triangle inequalities and the sine law relationship through investigations using dynamic geometry software or paper and pencil techniques. Accurate measurements of angles and sides are critical to the conclusions of the investigation. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems; - a collaborative contributor who works effectively as an interdependent team member. Strand(s): Trigonometry Unit 4 - Page 14 Principles of Mathematics - Academic Overall Expectations TRV.03D - solve trigonometric problems involving acute triangles. Specific Expectations TR3.01D - determine, through investigation, the relationships between the angles and sides in acute triangles using dynamic geometry software; TR3.02D - calculate the measures of sides and angles in acute triangles using the sine law and cosine law. Prior Knowledge properties of similar triangles; use of The Geometer’s Sketchpad™ Planning Notes Reserve computer lab time and use The Geometer’s Sketchpad™ to investigate acute triangles. Paper, pencils, rulers and protractors will be required if technology is not available. Use examples of acute triangles only. Teaching/Learning Strategies Teacher Facilitation: Review how to label a triangle using the upper and lower case letters prior to the activity. Uppercase letters are used for angles and lowercase letters for sides. For example, the side across from angle “A” is labelled side “a”. The teacher should monitor and aid those students who are experiencing difficulties during the activity. Student Activity: Students work individually or in pairs to complete the following investigation. Sines, Sines Everywhere Sines 1. Use The Geometer’s Sketchpad™ to draw any scalene acute triangle: Use the point tool to make three different points. Select any two points and choose Construct, Segment. A line segment will join the two points. Repeat to construct the other sides of the triangle. 2. Measure the degree measurement of all three angles and all three side lengths. 3. Arrange the measurements of the angles from greatest to least. 4. Arrange the measurements of the side lengths from longest to shortest. 5. Is there a relationship between the size of the angles and the length of the sides of the triangle? 6. What conclusions can be drawn about the size of an angle in a triangle and the length of the side opposite to it? 7. Use the Sketchpad’s calculator or your own calculator to calculate the sine ratio of each angle. 8. Create ratios comparing the sine of each angle with the side length across from the angle. 9. What do you notice about all these ratios? 10. Drag one of the vertices of the triangle to create another scalene triangle. Repeat steps 2 to 9 and determine if you have the same relationships. 11. Write a concluding statement about the two relationships you have discovered about triangles. Teacher Facilitation: After the activity, discuss the conclusions from the investigation. Lead the students in summarizing the Sine Law as a proportional statement. The reciprocal statement could also be shown. The teacher will demonstrate various examples of solving for sides and angles using the Sine Law. Be sure to include an example where the Sine Law cannot be used and discuss the requirements that are necessary to use the Sine Law, i.e., an angle and its opposite side must be known in order to make use of the Sine Law. Use the lesson to practise the mechanics of working with the sine law before incorporating applications of Sine Law. Unit 4 - Page 15 Principles of Mathematics - Academic Assessment/Evaluation Techniques A paper and pencil quiz could assess the student’s knowledge and understanding of how to use the Sine Law to solve acute triangles. Communication and Understanding can be assessed by conferencing with students on the relationships discovered during the activity. Independent Work skills could be assessed using a rubric for students working independently. Follow-up Skills: 300 minutes TR3.02 - calculate the measures of sides and angles in acute triangles, using the sine law and cosine law; TR3.03 - describe the conditions under which the sine law or the cosine law should be used in a problem; TR3.04 - solve problems involving the measures of sides and angles in acute triangles. Be sure to solve overlapping triangles with both sine law and the trig ratios (e.g., height of acute triangle). Extensions The Cosine law could also be treated as a special case of the Pythagorean theorem. In this case the contained acute angle is 90°. Therefore the expression of -2abcosC of the cosine equation will have a value of zero resulting in the equation of c2 = a2 + b2. Treatment of the Sine law is limited to acute triangles only in the Grade 10 course. However if time permits the Sine law could be extended to include obtuse triangles. When using obtuse triangles the ambiguous case should be addressed. Activity 4.6: Touring With Trigonometry Time: 75 minutes Description Students will use trigonometry to calculate lengths and angles of inaccessible objects that can be found in the schoolyard. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a collaborative contributor who works effectively as an interdependent team member; - a responsible citizen who respects the environment and uses resources wisely; - a collaborative contributor who exercises Christian leadership in the achievement of individual and group goals. Strand(s): Trigonometry Overall Expectations TRV.02 - solve trigonometric problems involving right triangles; TRV.03 - solve trigonometric problems involving acute triangles. Specific Expectations TR2.02 - solve problems involving the measures of sides and angles in right triangles; TR2.03 - determine the height of an inaccessible object in the environment around the school; TR3.04 - solve problems involving the measures of sides and angles in acute triangles. Unit 4 - Page 16 Principles of Mathematics - Academic Planning Notes Some of the outdoor activities require setup before the students begin the activity, e.g., a marker must be set in place for the soccer goal problem. Determine the inaccessible objects beforehand. Each group of students will need a clinometer to measure the angle of elevation of the inaccessible objects. Clinometers may be purchased, but making a clinometer is a beneficial and simple activity for the students. (Instructions are found later in this activity.) Students should master the use of the clinometer before attempting the field activity. Measuring tapes and/or trundle wheels are needed. (Pacing could be used, but accuracy will be affected.) Teachers should adapt the activities to suit their own individual schoolyards. A sample set of activities is included at the end of this description. Prior Learning Required trigonometric ratios for right-angled triangles, sine law, cosine law Teaching/Learning Strategies Teacher Facilitation: Set the scenario where a person needs to calculate the height of an inaccessible object. Discuss how surveyors use transits to measure angles and distances in a three-dimensional setting. Provide instruction on how to construct and use a homemade clinometer to model a transit. Describe the location of the outdoor objects whose measures are to be determined. Student Activity Working in groups of three or four, the students will measure the required lengths and determine angles of elevation in the vertical plane, and angles on the horizontal plane. They will record the required data on a diagram, return to the classroom, and then solve the problems. A complete set of solutions will be prepared and submitted by each individual student. Assessment/Evaluation Techniques As the students work on the field activity, the teacher could make observations on Learning Skills, specifically teamwork and initiative. The activity provides an opportunity for formative assessment to determine areas where students do not have a firm understanding of trigonometry and need more assistance. For example, difficulty with trigonometry terminology and weak technology skills will become evident as the students prepare their written reports. Unit 4 - Page 17 Principles of Mathematics - Academic How to make a Clinometer Wooden chalkboard protractors are easily converted into clinometers. (Another possibility is to glue an enlarged photocopy of a protractor onto a piece of cardboard.) Tape a straw onto the flat side of the protractor. The straw will serve as a viewfinder. Attach a weight (e.g., washer, bolt) to the end of a string. Tape the string to the midpoint of the straight edge on the protractor. To determine an angle of elevation, gaze through the straw to look at the top of the examined object. The weight will hang perpendicularly to the ground. As one student views through the straw, another student examines the angle marked by the weighted string. The string indicates the measure of the complementary angle to the angle of elevation. Follow-up Skills: 75 minutes Practise solving triangles that use combinations of all the trigonometric methods studied thus far. Unit 4 - Page 18 Principles of Mathematics - Academic Trigonometry Field Trip Record the data necessary to determine the following measures. Return to the classroom to complete the calculations. 1. Determine the height of the tree. 2. Determine the height of the flagpole 3. A soccer player attempts to kick a goal from the location marked on the soccer field. Measure the three distances labelled on the diagram and then calculate the angle within which the kick must be made. 4. Determine the height of the school crucifix without using the trig ratios of right-angled triangles. 5. Determine the length of the school. You may use a tape measure for one measurement only. Unit 4 - Page 19 Principles of Mathematics - Academic Activity 4.7: Trigs of the Trade Time: 150 minutes Description Students describe the application of trigonometry by researching a career or application of trigonometry in everyday life. They will write a report, design a display and then make an oral presentation on their research project during a class Math Fair. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - a self directed, responsible, life long learner who sets appropriate goals and priorities in school, work and personal life; - an effective communicator who 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 self directed, responsible, life long learner who examines and reflects on one’s personal values, abilities and aspirations influencing life’s choices and opportunities. Strand(s): Trigonometry Overall Expectations TRV.02 - solve trigonometric problems involving right triangles; TRV.03 - solve trigonometric problems involving acute triangles. Specific Expectations TR3.05 - describe the application of trigonometry in science or industry; TR2.02 - solve problems involving the measures of sides and angles in right triangles; TR3.04 - solve problems involving the measures of sides and angles in acute triangles. Planning Notes Reserve Library/Resource Centre and computer lab time. Ensure that resources are adequate for researching trigonometric applications (e.g., Internet access, books, magazines, math textbooks). A Math Fair is an activity where students take turns presenting their projects as displays to each other or to another class. Teachers may choose to alter the format, having students prepare a written report and oral presentation to the full class instead of preparing a display. Teaching/Learning Strategies Teacher Facilitation: Lead a full class brainstorming session to discuss and list careers and everyday applications of trigonometry. Compile a list of potential research topics during this discussion. Describe the Math Fair set up and the expectations for the research paper or display. (If students cannot suggest a variety of careers and applications, the topics listed below could be placed on pieces of paper and each student could draw a topic out of a hat and then research the topic selected.) Student Activity Students research a career or application individually or in pairs. They prepare a report or poster display on their topic. The report must include at least one sample problem that illustrates an application and solution involving the use of trigonometry. A brief oral presentation will be made to accompany the display at the Math Fair. Some suggested fields in which trigonometry is used: Automotive Industry - tool and die design, robotics Forensic Science - crime scene investigation, accident site reconstruction Unit 4 - Page 20 Principles of Mathematics - Academic Medical Fields - ultrasound, radiation therapy Building Construction - extensive use of triangles in roof rafter construction, supports Aviation - calculating height of cloud cover, navigational equipment, angles of ascent and descent, electronics, positioning used by flight computers Astronomy - calculating distance in space, size of planets Geology - locate rocks below earth’s surface, draw maps in three-dimensional space Architecture/Drafting - designing buildings, blueprints, designing awnings based on sun’s angle to admit sun in winter but prevent sunshine in summer Engineering - designing bridges, dams, overpasses, roads, underground locations of sewer pipes GPS - using 3-D trigonometry to determine the Global Position, locate satellites, give people directions Armed Forces - radio contact, map reading and design, missile deployment Navigation - lighthouses, rescue teams, watchtowers Surveying - map making, boundary measurement Design Art - geometric designs for artwork, flooring, windows, quilting Agriculture - drainage systems, laser-controlled devices Science - Snell’s Law, Optics, Electronic, Statics Geography - contour maps Assessment/Evaluation Techniques To ensure that all students are aware of the project expectations and assessment format, teacher and students could develop a rubric for assessing the activity as part of the discussion that takes place before beginning the activity. A sample rubric has been designed specifically for this activity, as a reference. Because of the richness of this activity, assessment is possible in all four categories of the Achievement Chart. Learning Skills could be assessed, specifically initiative, throughout the project. If the projects are presented using a Math Fair approach, peer assessments could be completed by the students, using a checklist rating scheme. Teachers may wish to videotape the presentations to use as exemplars in the future. Resources Information on careers in mathematics and related sample problems can be found at these web sites: www.coolmath.com/careers.htm http://on.cx.bridges.com Unit 4 - Page 21 Principles of Mathematics - Academic Rubric Assessing “Trigs of the Trade” Categories Communication Description of Career Sources Explanation of Trigonometric Connections Effectiveness of Display Application Problem(s) and Solution(s) Pertaining to Chosen Career Level 1 50-59% Level 2 60-69% Level 3 70-79% - limited use of resources - some use of resources - use of a variety of resources - limited detail and clarity - some detail, stated clearly sometimes - considerable detail, most often stated clearly Level 4 80-100% - employed extensive and varied resources - extensive detail, consistently stated clearly - chooses - chooses - chooses - chooses an appropriate appropriate appropriate extensive variety materials with materials with materials of appropriate considerable some teacher independently materials teacher assistance assistance independently Appropriateness - problem is - problem is - problem is - problem is of Posed Problem related to chosen related to chosen related directly to related directly to career in limited career to some the career chosen, chosen profession degree, limited degree, some use frequent use of with consistent use of appropriate of appropriate appropriate use of appropriate terminology terminology terminology terminology Solution(s)’ - minimal - some - adequate - full justification, Logical justification, justification, justification, correct and Progression solution contains solution contains correct solution elegant solutions significant errors minor errors Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity. Unit 4 - Page 22 Principles of Mathematics - Academic Peer Evaluation of Math Fair Display and Presentation Not Able to Determine Needs Improvement Satisfactory Good Excellent Knowledge of the topic (well researched, explained and understood) Problem Solving (presentation, understanding, solution and discussion of the trigonometric problem) Communication (use of mathematical language, eye contact, enthusiasm, answering of questions) Initiative (appearance of display, effort in preparing display, creativity) Unit 4 - Page 23 Principles of Mathematics - Academic Unit 5: Summative Assessment Time: 8 hours Unit Description This unit is made up of a series of performance tasks in which students will need to use all the knowledge and understanding of content and procedures of this course. The activities are based on the central theme of an amusement park. Teachers should choose the activities that will address as many of the expectations of the course as possible and still fit within their own time scheme. Any extra time should be used for other forms of examination review. Students also write a formal paper and pencil final exam. Note: Some of these activities may be used as final assessment instruments, final assessment review activities, or diagnostic tools. Teachers should combine a mixture of these activities along with a formal written exam in order to provide a comprehensive evaluation package. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be an effective communicator who: - reads, understands, and uses written materials effectively; - presents information and ideas clearly and honestly and with sensitivity to others. The graduate is expected to be a reflective and creative thinker who: - creates, adapts, evaluates new ideas in light of the common good; - thinks reflectively and creatively to evaluate situations and solve problems; - adapts a holistic approach to life by integrating learning from various subject areas and experience. The graduate is expected to be a self-directed, responsible, life long learner who: - applies effective communication, decision-making, problem-solving, time and resource management skills. The graduate is expected to be a collaborative contributor who: - works effectively as an interdependent team member; - achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others. The graduate is expected to be a responsible citizen who: - accepts accountability for one’s own actions. Strand(s): Quadratic Functions, Analytic Geometry, Trigonometry Overall Expectations: All from the Quadratic Functions Strand All from the Analytic Geometry Strand All from the Trigonometry Strand Specific Expectations: All from the Quadratic Functions Strand All from the Analytic Geometry Strand All from the Trigonometry Strand Activity Titles (Time + Sequence) The following table provides a list of the activities that can be used as part of the performance component of the final summative assessment. The number of activities done, and the order in which they are done, is subject to the discretion of the teacher, based primarily on time constraints. Every attempt should be made to schedule activities that address as many of the course expectations as possible within the teacher’s own time scheme. Unit 5 - Page 1 Principles of Mathematics - Academic Activity 5.1 Activity 5.2 Activity 5.3 Activity 5.4 Ball-Is-Tic Use quadratic functions to determine the dimensions of a ball tossing range for the arcade area Survey Use proportional reasoning skills to investigate mathematical principles within the context of an amusement park 75 minutes Roller Coaster Use a variety of geometric (and algebraic) principles to construct a roller coaster Treasures of Math (Diagnostic Review) Use mathematical clues that address many of the course expectations in order to find a buried treasure 150 minutes 150 minutes 105 minutes Prior Knowledge Required All course expectations Use of graphing calculators and computers (spreadsheets, dynamic geometry software, and graphing software) Unit Planning Notes Timing: The activities allow teachers to choose appropriate activities that will fit into their time scheme and at the same time address as many of the course expectations as possible. Students will need graph paper and mathematical sets. A class set of graphing calculators (or computer software) and/or dynamic geometry software (e.g., The Geometer’s Sketchpad™) should be made available although they are not absolutely necessary for most activities. Use of CAD software, even though it is not necessary, is encouraged for those students (and teachers) who feel comfortable with it for the Roller Coaster activity. Students must be informed, in advance, of the type of assessment that will be used (verbal presentation rubric, written report rubric, etc.). Activities 5.2 and 5.4 may be used for course review purposes or diagnostic assessment purposes. Teachers should also supplement these activities with exercises from the student textbook and/or other resources. Teaching/Learning Strategies As throughout the course, this unit suggests a balance of small group and individual performance tasks through student-centred and teacher-directed activities. The activities, centred on the theme of an amusement park, provide the student and teacher, with the opportunity to assess the student’s overall achievement of the course’s expectations. The use of technological tools and software is highly recommended. Assessment and Evaluation A balance of assessment tools and strategies are recommended, even though the most prevalent one should be the Written Report Rubric (use Appendix D as a model). Careful consideration should always be given to addressing all four categories of the Achievement Chart in a planned and deliberate manner. If Activity 5.2 and/or Activity 5.4 are used for diagnostic purposes, teachers must allow sufficient time for feedback and student modification. Unit 5 - Page 2 Principles of Mathematics - Academic Resources Spreadsheets Graphing software The Geometer’s Sketchpad™ (or other dynamic geometry software) Computer-Aided Drafting software (CAD) Internet (Numerous sites exist. Search using keywords such as “mathematics”, “amusement park,” and “roller coaster”) Activity 5.1: Ball-Is-Tic Time: 150 minutes Description Students will use their knowledge and understanding of parabolic functions to determine the design of a ball tossing range in the arcade of an amusement park. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be - an effective communicator who reads, understands, and uses written materials effectively; - an effective communicator who presents information and ideas clearly and honestly and with sensitivity to others; - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and solve problems. Strand(s): Quadratic Functions Overall Expectations: All in the Quadratic Function Specific Expectations: All in the Quadratic Function Planning Notes Students will need graph paper and a graphing calculator or a computer with graphing software. Prior Knowledge Required all expectations in the Quadratic Function Strand use of scientific calculator or computers with graphing software Student Activity The amusement park asks you to design a ball throwing range for an arcade game. You throw a ball several times and determine the following design parameters. (a) Over a flat range, if a ball reaches a maximum height of 6 metres, you will hit a target that is 20 metres from where you throw the ball. What equation will describe this trajectory? Teacher Facilitation: Astute students will include the height at which the ball is released above the ground, due to a person’s height. This increased complexity will help establish student skill levels. (b) How would the equation change if the trajectory had the same initial and terminal points, but less arc? (a decreases as k decreases, but h remains the same) (c) To make the game more challenging you decide to have the ground slope upward from the throwing zone to the target. The slope will be 0.3. Using a graphing calculator determine how far from home plate the ball will land. Unit 5 - Page 3 Principles of Mathematics - Academic Teacher Facilitation: This portion of the activity will allow students to apply their knowledge and provide an opportunity for Level 4 achievement. The teacher may challenge the class for an algebraic solution which will require calculation of the intersection of a linear and a quadratic function. This requires strong Level 4 understanding. Assessment/Evaluation Techniques A written report rubric (use Appendix D as a model) may be used for the report. If the teacher wishes to have presentations instead of a written report, verbal presentation rubric (Appendix C) may be used, using criteria similar to that in the Written Report Rubric below. The teacher may wish to assess the use of graphing calculators using the Observation Rubric. Specifically, the Written Report Rubric is shown below using the criteria for this activity. Subsequent activities may use this as a template to be adapted to their own specific criteria. Written Report Rubric: Ball-Is-Tic Category Level 1 Level 2 Level 3 Level 4 50-59% 60-69% 70-79% 80-100% - limited ability - moderate - correctly draws - accurately Knowledge and with graphs and ability with graph and graphs and Understanding Determines the equations and/or graphs or expresses the expresses the connection between the requires equations equation in equations and graph and the equation frequent proper algebraic includes varying of the quadratic function assistance by form initial height (Part A) others differences - minimal - identifies - identifies - correctly Problem Solving, understanding or changes with changes in a, h, identifies and Thinking and Inquiry Explain the role of a, h, limited some major and k with minor explains the roles and k in the graph of explanation errors errors of a, h, and k y = a(x – h)2 + k (Part B) - limited use of - moderate use - considerable - excellent use of Written quadratic of quadratic use of quadratic quadratic Communication Uses terminology terminology terminology terminology quadratic terminology - attempts to - error in one - finds point of - finds point of Application Applies skills of find point of equation or intersection intersection quadratic equations, intersection with point of accurately accurately, both linear equations and incorrect intersection graphically and graphing calculator to equations algebraically solve a problem (Part C) Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity. Accommodations The teacher prompts students who may be experiencing difficulties. Unit 5 - Page 4 Principles of Mathematics - Academic Activity 5.2: Survey Time: 150 minutes Description In this activity students will use their knowledge and understanding of trigonometry and Analytic Geometry to investigate some design features of an amusement park. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be: - an effective communicator who reads, understands, and uses written materials effectively; - a reflective and creative thinker who adapts a holistic approach to life by integrating learning from various subject areas and experience; - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills; - a responsible citizen who accepts accountability for one’s own actions. Strand(s): Trigonometry, Analytic Geometry Overall Expectations: All in the Trigonometry and Analytic Geometry Strand Specific Expectations: All in the Trigonometry and Analytic Geometry Strand Planning Notes Students should be able to do this activity individually. Teacher should inform students in advance as to whether a written report or a verbal presentation will be assessed and evaluated. Teachers may wish to add their own additions and variations to the following activity. Students will need graph paper and mathematical set. Scientific calculators (or computers) should be made available. Expectations that are not explicitly referenced in this activity should be supplemented by exercises from the textbook or other sources. Prior Knowledge Required all expectations in the Trigonometry Strand use of scientific calculator (or computers) Teaching/Learning Strategies Teacher Facilitation: Present the following scenario to the class: Student Activity A portion of a theme park is located on a perfectly square parcel of land, two km per side. A 70 m high escarpment forms the west boundary. Several mathematical principles have been incorporated in the design of this park. Your role is to analyse and describe the mathematical principles involved and present your results in a report. 1. For reference purposes, draw a map of the parcel of land on graph paper. Establish a suitable grid and scale. Position the origin at the centre. State the scale you are using. 2. A circular ring road passes around the park and touches the boundary exactly at the midpoint of each side of the square park. Each of these is an exit point for this section of the park. Unit 5 - Page 5 Principles of Mathematics - Academic Teacher Facilitation: If necessary, direct the class towards a diagram similar to the following: 3. Accurately draw the ring road on your map and label the exits according to their direction of opening. N, E, S, or W) 4. Determine the equation of the circle that describes the ring road. 5. Six straight roads directly connect each pair of exits. Draw each of them on your map. With respect to the circle on your map, what is the mathematical term for each of the lines representing these six roads? 6. How would you use analytic geometry to determine the lengths of each of these six roads, using the coordinates on your map? Use this method to calculate the lengths of the roads. 7. Calculate the coordinates of the midpoints of the chords NE, ES, SW, and WN. Label them P, Q, R, and S respectively (e.g., The midpoint of NE is P, etc.). 8. Draw four roads, each passing from the origin to the midpoint of each of the diagonal roads NE, ES, SW, and WN. Determine the angle of intersection of each of these roads with respect to the others. Verify your conjecture using congruency, slopes and/or trigonometry. 9. In the resulting diagram, there are many properties and relationships that you learned in the course. You are to list and describe characteristics relating to: Angles relating to parallel lines Altitudes, perpendicular bisectors, medians, etc. Diagonals of a quadrilateral Teacher Facilitation: Teachers may wish to offer students ample direction that ensures that they review important expectations, or they may wish to use supplementary exercises from the text. 1. A cable car travels from park-level to the top of the escarpment. From the park platform, you measure a 60° angle of elevation to the upper escarpment platform. 2. Use trigonometry to determine the distance from the lower platform (L) to the point (B) directly below the upper platform (E) (Note: That point will likely be inaccessible at this time, inside the base of the escarpment.) 3. If you were located at the lower platform, how would you use a metre stick and your knowledge of similar triangles to estimate the distance from the platform (L) to the point B. Unit 5 - Page 6 Principles of Mathematics - Academic 4. On your map, mark a possible location for the the base platform (L). 5. Three cables will be required to transport the cable car to the top terminal. Determine the total length of cable required. Add an additional 30 m for tie-down purposes. Assessment/Evaluation Techniques A Written Report Rubric (Appendix D) can be used on the written report. If the teacher wishes to have presentations instead of a written report, Verbal Presentation Rubric (Appendix C) can be used. Knowledge can be assessed using a marking scheme and the following skills: calculation of midpoints and length of line segments, equation of circle, characteristics of a quadrilateral, measurement of sides with right angle trigonometry, law of sines or cosines. Assess Problem Solving Communication, and Application by adapting the rubric illustrated in Activity 5.1 using the following criteria: Assess Communication using AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solution Assess Problem Solving using AG2.03 - solve multi-step problems, using the concepts of slope, the length, and the midpoint of line segments Accommodations The teacher should be available to prompt students who may be experiencing difficulties. Activity 5.3: Roller Coaster Time: 150 minutes Description In this activity students will design a roller coaster that will demonstrate their understanding of the quadratic function, trigonometry, and analytic geometry. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be an effective communicator who: - reads, understands, and uses written materials effectively; - presents information and ideas clearly and honestly and with sensitivity to others. The graduate is expected to be a reflective and creative thinker who: - creates, adapts, evaluates new ideas in light of the common good; - adapts a holistic approach to life by integrating learning from various subject areas and experience. The graduate is expected to be: - a self-directed, responsible, life long learner who applies effective communication, decision-making, problem-solving, time and resource management skills. The graduate is expected to be a collaborative contributor who: - works effectively as an interdependent team member; - achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others. The graduate is expected to be a responsible citizen who/; - accepts accountability for one’s own actions. Strand(s): Quadratic Functions, Analytic Geometry, Trigonometry Unit 5 - Page 7 Principles of Mathematics - Academic Overall Expectations: All from the Quadratic Functions Strand All from the Analytic Geometry Strand All from the Trigonometry Strand Specific Expectations: All from the Quadratic Functions Strand All from the Analytic Geometry Strand All from the Trigonometry Strand Planning Notes Students will need graph paper and mathematical sets. Students will work in groups of two or three. Students who have knowledge and/or expertise with CAD (or similar software) should be encouraged to use it along with their hand drawn model. The teacher should have visual models available for students to view (from Canada’s Wonderland, Cedar Point, various Internet sites, etc.). Students will be expected to show appropriate “theoretical” work associated with the various shapes on the roller coaster (i.e., find equations of the various shapes). Prior Knowledge Required all course expectations Teaching/Learning Strategies Teacher Facilitation: Students should be put in groups of two or three. Students should be shown some visual models of various roller coasters. (These can be obtained from brochures of Canada’s Wonderland, Cedar Point, Internet sites, etc.) Students may need explanations with some of the instructions in the student worksheet (e.g., the “sample” problem in question 5 of Part A refers to the trigonometric approach to finding the height of tall object). Student Worksheet Part A: Your task is to design a roller coaster according to the following criteria: 1. The site on which the roller coaster is to be built is flat and has dimensions 200 m by 500 m. 2. The maximum height of any part of the roller coaster is 30 m. 3. All the sections of the track must be circular, parabolic, or rectilinear. These sections must be as seamless as possible. 4. The roller coaster must include at least three “hills” made up of parabolas, two “loop-the-loops” made up of circles and one straight section. 5. The site contains several trees and the management wishes to preserve as many of them as possible. All three of the “hills” must be built over trees. Part of your task is to determine a trigonometric method of determining the height of the trees in order to design the roller coaster. Be sure to include a “sample” problem to illustrate your method. Unit 5 - Page 8 Principles of Mathematics - Academic 6. The last section of the track must be straight and horizontal. This section must be long enough to allow the carts to come to a safe stop. Graphing the following data and then interpolating or extrapolating can be used to estimate the speed of the carts at the bottom of the ride. Max. Height – Height of last section (m) Initial speed on last section (m/s) 5 8.9 10 12.1 15 15.1 20 16.9 25 19.3 30 21.1 35 22.5 40 24.1 45 25.9 50 27.1 The braking mechanisms available provide the following stopping distances given the speeds at the beginning of the horizontal section. Speed (m/s) Stopping Distance (m) 0 0 2 0.2 5 1.3 8 3.2 10 5.1 13 8.6 16 12.6 18 16.3 20 19.8 22 24.3 Construct a graph of the above data. Use the graph to interpolate or extrapolate in order to find the required length of the last section of the track in your roller coaster. 7. Your product must include: i) a scale diagram of the site indicating the location of the roller coaster; ii) a labelled graph showing all sections of the roller coaster; iii) the theoretical work done to select the equations of each section of the roller coaster. Assessment/Evaluation Techniques Use the rubric from Activity 5.1 as a model to assess the following criteria: Assess Inquiry and Problem Solving using the expectation QF4.02 - determine the zeros and the maximum and minimum values of a quadratic function, using algebraic techniques Assess Communication using the expectation AG2.04 - communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions Assess Application using the expectation AF4.03 - solve problems related to an application, given the graph or the formula of a quadratic function Unit 5 - Page 9 Principles of Mathematics - Academic Knowledge could be assessed using a marking scheme with attention to these topics: graphing techniques, interpreting data and graphs, use of equations of lines, circles and parabola, use of trigonometry to solve problems. Building the physical model of the roller coaster is not an expectation and should not be assessed. However, when assessing Learning Skills (Rubric Appendix A) the student’s initiative should be noted. Other Learning Skills that could be assessed during this activity are Teamwork and Organization. Activity 5.4: Treasures of Math (Diagnostic Review) Time: 105 minutes Description In this activity students will use their knowledge and understanding of the content and procedures of this course to find a buried treasure in the sandpit on Pirate’s Island with a variety of mathematical clues. Although the activity could be used as a summative assessment tool, it is better suited as a diagnostic tool to indicate areas needing review for the formal exam. It can be used in place of, or in addition to, exam review material taken from the more traditional textbook source. Strand(s) and Expectations Ontario Catholic School Graduate Expectations The graduate is expected to be an effective communicator who: - reads, understands, and uses written materials effectively; - presents information and ideas clearly and honestly and with sensitivity to others. The graduate is expected to be a reflective and creative thinker who: - creates, adapts, evaluates new ideas in light of the common good. The graduate is expected to be a self-directed, responsible, life long learner who: - applies effective communication, decision-making, problem-solving, time and resource management skills. The graduate is expected to be a collaborative contributor who: - works effectively as an interdependent team member; - achieves excellence, originality, and integrity in one’s own work and supports these qualities in the work of others. - The graduate is expected to be a responsible citizen who accepts accountability for one’s own actions. Strand(s): Quadratic functions, Analytic Geometry, Trigonometry Overall Expectations: All from the Quadratic Functions Strand All from the Analytic Geometry Strand All from the Trigonometry Strand Specific Expectations: All from the Quadratic Functions Strand All from the Analytic Geometry Strand All from the Trigonometry Strand Planning Notes Students will need graph paper and mathematical sets. Teachers must have enough of the clues photocopied on separate numbered pieces of paper. Graphing calculators (or computer software) and/or dynamic geometry software may be beneficial for some of the clues, but not absolutely necessary. Students can be encouraged to develop their own treasure map and clues. Unit 5 - Page 10 Principles of Mathematics - Academic Prior Knowledge Required all course expectations use of graphing calculator (or computer software) and/or dynamic geometry software Teaching/Learning Strategies Teacher Facilitation: Students should be put in groups of two or three. All students are expected to hand in individual reports. The teacher must have clues photocopied (and numbered) for all groups. Students must present a well-organized solution to any clue before the next one is given to them. Students set up a grid on graph paper (use centimetre graph paper or larger). The teacher must be aware that the graph could get congested (urge students to make a large enough grid). Encourage students to use coloured pencils to illustrate the many different items that will appear on the graph paper. Student Activity On Pirate’s Island in the amusement park is a square sand pit that is 20 m by 20 m. It has a grid of ropes that are 1 m apart both horizontally and vertically. There is a treasure buried somewhere in the sand pit. Use the clues to find where the treasure is buried and what the treasure is. On a large enough piece of graph paper, construct the coordinate axes to model the 20 x 20 sand pit. Put the origin, O (0, 0), at the centre of the pit. Clue 1 Using a centre of (0, 0) draw a circle of radius 10 on your graph paper. Find two points on the circle that have integer coordinates (and neither of these points can have the same x- or y-coordinates). Label these two points as A and B. Find the equation of this circle. Verify that points A and B are on the circle. Teacher Facilitation: Students should be choosing points such as A(6, 8) B(-8, 6) or A(-6, -8) B(8, -6). Direct students, if necessary, to choose points so that AB is a chord, not a diameter. Ensure that students have shown the necessary work before giving them Clue 2. Clue 2 Find the equation of the perpendicular bisector (b1) of AB. Verify that the centre of the circle lies on the perpendicular bisector. Find length of the line segment (L1) on the perpendicular bisector that has its endpoints on the circle. (Let this number be k.) Teacher Facilitation: Students should get k = 20. Ensure that students have shown the necessary work before giving them Clue 3. Clue 3 Draw a line segment (L2) from the centre of the circle to the point C(k, 0). Verify that point C is on the circle. Draw a line segment (L3), with one endpoint at (0, 0) and the other endpoint on the circle, in Quadrant IV that makes an angle of 30° with L1. Find the exact coordinates of the point where L3 intersects the circle (call this point D). Teacher Facilitation: In order for students to find the coordinates of point D, the teacher may have to prompt the student to draw a perpendicular from point D to the x axis, then use trigonometric ratios. Ensure that students have shown the necessary work before giving them Clue 4 Unit 5 - Page 11 Principles of Mathematics - Academic Clue 4 Find the zeros and vertex of the parabola y = x2 + 4x – 12. (Label these points P, Q, R.) Find a fourth point that could be used with the zeros and vertex of the parabola to form a parallelogram. (Label this point S.) Find the intersection point of the diagonals of the parallelogram. (Call this point E.) Teacher Facilitation: Points P, Q, R are P(-6, 0) Q(2, 0) R(-2, -12) (not necessarily in this order). There are several possibilities for point S (such as S (6, -12); S(-2, 12)). These in turn mean several possibilities for point E. Ensure that students have shown the necessary work before giving them Clue 5. Clue 5 The buried treasure is at the midpoint of line segment DE. It is a special millennium coin that is worth $100 times the length of the diagonals in Clue 4. Where is the gold coin at and how much is it worth? Teacher Facilitation: Since there were several possibilities for point E there will be several possible “final” answers. Encourage students to make their own treasure hunt that will incorporate as much content and as many procedures as possible that was covered in the course. (This could be done by students individually or in small groups.) Assessment/Evaluation Techniques Since this activity is designed as a diagnostic review for the final exam, a formal assessment should not be done by the teacher. Each student should conduct a self-evaluation of skills needing review after the activity has been completed and corrected. These skills should be found in textbook exercises and practised by the student. Accommodations The teacher should be available to prompt students who may be experiencing difficulties, especially to ensure a “correct” response and appropriate work shown before students are allowed to move on to the next clue. Unit 5 - Page 12 Principles of Mathematics - Academic Appendices Appendix A Learning Skills Works Independently: Needs Improvement - works with ongoing and regular supervision - selects materials, resources and activities with direction and assistance - persists with tasks with help and encouragement - explores and/or selects and/or uses a variety of learning strategies with direction and assistance The student requires no supervision and is self-reliant Satisfactory Good Excellent - works with moderate - works with minimal - works without supervision supervision supervision - selects materials, - selects materials, - selects materials, resources and activities resources and activities resources and activities with assistance with minimal independently supervision - persists with tasks - persists with tasks - persists with tasks when encouraged with minimal independently supervision - explores and/or - explores and/or - explores, selects and selects and/or uses a selects and/or uses a uses a variety of variety of learning variety of learning learning strategies strategies when strategies independently and provided with effectively assistance Work Habits/Homework: The student can demonstrate homework completion, on-task behavior, and appropriate classroom behavior. Needs Improvement - limited homework completion - follows instructions with considerable supervision Satisfactory - completes homework with frequent reminders - follows instructions with supervision - rarely takes complete or accurate notes - occasionally takes complete and accurate notes - accepts responsibility for work completion and behavior after conferencing takes place - accepts responsibility for work completion and behavior with intervention Page i Good - regularly completes homework with few reminders - follows instructions accurately with a minimum of supervision - regularly takes complete and accurate notes - accepts responsibility for work completion and behavior Excellent - completes homework consistently - follows instructions accurately and without supervision - consistently takes complete accurate notes and adds own annotations - accepts responsibility and consequence for work completion and behavior Principles of Mathematics - Academic Appendix A (Continued) Learning Skills Teamwork: The student will openly contribute and commit, share information, develop ideas, and show respect to overall group effort. Needs Improvement Satisfactory Good Excellent - limited ability to work - occasionally - usually identifies - consistently identifies toward group goals identifies and works group goals and works group goals and works toward group goals to meet them diligently to meet them - limited ability to - occasionally shares - usually shares - consistently shares share information or information or follows information and information and follow direction direction follows direction accurately follows directions - rarely performs more - occasionally performs - usually performs - consistently performs than one role more than one role more than one role more than one role - seldom expresses - occasionally - usually expresses - consistently promotes ideas within a group expresses ideas within ideas and interacts group interaction a group within a group Organization: The student has the ability to give structure or order to a task or process. Needs Improvement Satisfactory Good Excellent - follows established - follows established - follows established - follows established routines with daily routines with reminders routines and develops routines and develops supervision personal routines and excellent personal consistently follows routines and follows both both independently and with a high degree of success - limited ability to meet - occasionally meets - regularly meets - consistently meets deadlines deadlines deadlines deadlines - rarely brings - occasionally brings - usually brings - consistently brings resources needed for resources needed for resources needed for resources needed for class class class consistently class brings resources needed for class - rarely organizes time - occasionally - usually organizes - consistently organizes organizes time time time Page ii Principles of Mathematics - Academic Appendix A (Continued) Learning Skills Initiative: The student can be a self-starter. The student begins tasks immediately, demonstrates leadership skills, actively promotes group interaction, creates new ideas, mentors and assists others, is creative and enthusiastic, is a confident learner and accepts responsibility. Needs Improvement Satisfactory Good Excellent - rarely anticipates - occasionally - usually anticipates - consistently tasks and needs anticipates tasks and tasks to be done and anticipates tasks and supervision to start and frequently needs moves to complete immediately and complete them supervision to start them independently moves to them complete them - developing ability to - admissible ability to - usually makes a plan - consistently makes a make a plan and needs make a plan and needs and can move to follow plan and independently strategies to follow supervision to follow through follows through through through - seldom leads group - occasionally makes - frequently leads - accepts authority and interaction attempts to show group interaction leads group interaction leadership in group interaction - seldom volunteers - volunteers creative - regularly volunteers - consistently suggests creative ideas ideas when asked creative ideas and creates new ideas opinion and solicits ideas from others Page iii Principles of Mathematics - Academic Appendix B Sample Observational Rubric Criteria Level 1 50-59% begins task but Engages in task with considerable prompting Applies appropriate strategies - requires consistent support and prompting to pursue alternative strategies Uses resource materials effectively and independently - does not refer to notes, text, or other resources before seeking assistance Works effectively with others in the group - assumes passive role and contributes infrequently and often in a limited way - contribution is limited and only when prompted Contributes effectively to the work of the group Level 2 60-69% - begins task with some prompting Level 3 70-79% - begins task without prompting - pursues alternative strategies with frequent assistance and limited prompting - rarely refers to notes, text, or other resources before seeking assistance - assumes passive role and contributes usually only when prompted - contribution is infrequent but sometimes will volunteer ideas - pursues alternative strategies with only limited assistance Uses the materials and resources effectively - requires frequent support in finding and using the materials - will often need support in finding or using the materials Is an active problem solver - will explore very few possibilities and often stops before it is solved - rarely seeks alternate solutions - will explore some possibilities but may stop before problem is solved - sometimes seeks alternate solutions Level 4 80-100% - begins task promptly and encourages others to begin - actively pursues alternative strategies independently - frequently refers to notes, text or other resources before seeking assistance - assumes active role and contributes freely to the group - consistently refers to notes, text or other resources before seeking assistance - assumes leadership role and tries to encourage all to contribute - contribution is frequent and does not require prompting to share ideas - usually needs only limited and infrequent support in finding or using the materials - contribution is frequent and builds on others ideas - will explore a few possibilities until the problem is solved - often seeks alternate solutions - consistently needs little or no support to find and use materials and consistently assists others to find and use - will explore many possibilities until the problem is solved - consistently seeks alternate solutions From the Principles of Mathematics Course Profile Grade 9, Catholic Curriculum Cooperative Writing Partnership Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity. Page iv Principles of Mathematics - Academic Appendix C Verbal Presentation Rubric (Achievement Chart Categories) Category Knowledge and Understanding Thinking, Inquiry and Problem Solving Level 1 50-59% - demonstrates a limited understanding of concepts - limited ability to perform algorithms accurately or can only perform simple algorithms - presents simple arguments Level 2 60-69% -demonstrates some understanding of concepts -performs algorithms with some accuracy Level 3 70-79% - demonstrates a considerable understanding of concepts - frequently performs algorithms accurately Level 4 80-100% - demonstrates a thorough understanding of concepts - consistently uses the most effective and efficient algorithm accurately - clearly presents arguments of moderate complexity both written and orally - some clarity - clearly presents valid arguments of considerable complexity both written and oral - routinely presents complex arguments both written and orally effectively - limited use of appropriate math language - limited effectiveness - inconsistent use of appropriate math language - some effectiveness - regular use of appropriate math language - effective presentation - ideas are seldom explained clearly - ideas are somewhat clearly explained - ideas are clearly explained - unable to answer class questions effectively - can answer few questions effectively - consistently use of appropriate math language - effective and polished presentation - ideas are clearly explained with examples and justifications - can answer most class questions effectively - arguments are often difficult to follow Communication 1. Language use 2. Presentation effectiveness 3. Presentation of Ideas and Answering of Questions - can answer the majority of class questions effectively limited can justify - justifies - generalizes Applications justification of relationships with relationships with relationships relationships prompting respect to the task Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity. Page v Principles of Mathematics - Academic Appendix C (Continued) Verbal Presentation Rubric (Learning Skills) Skills Organization Team Work Initiative (preparation) Level 1 50-59% - presentation is extremely choppy and many clarifications must be made - demonstrates only slight respect, leadership, cooperation, and participation Level 2 60-69% - presentation appeared choppy and some clarifications must be made - demonstrates moderate respect, leadership, cooperation, and participation Level 3 70-79% - organized and presentation flows sequentially Level 4 80-100% - well organized and presentation flows extremely well - demonstrates respect, leadership, cooperation, and participation - demonstrates respect, leadership, cooperation, and constant participation - presentation needs major modifications - presentation needs slight modifications - presentation is prepared - presentation is well prepared Adapted from York Region Catholic District School Board Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity. Page vi Principles of Mathematics - Academic Appendix D Written Reports Rubric (Achievement Chart Categories) Category Knowledge and Understanding Level 1 50-59% - demonstrates limited understanding of concepts - performs simple algorithms with considerable help - tables and graphs not present or inaccurate Thinking/Inquiry and Problem Solving - can only explain simple arguments - problem-solving steps are missing or presented in a cluttered, unclear manner - limited use of mathematical language - inappropriate or ineffective conclusions - limited application of concepts, procedures to problems seen in familiar settings Level 2 60-69% - demonstrates a somewhat clear understanding of concepts - algorithms performed with inconsistent accuracy - tables and graphs presented with inconsistent accuracy - explains arguments of moderate difficulty - applies problemsolving steps with a moderate level of clarity and understanding - satisfactory use of mathematical language - inconsistent justifications of conclusions - inconsistently applies concepts, procedures to problems seen in a familiar context Level 3 70-79% - demonstrates considerable understanding of concepts - algorithms performed accurately - tables and graphs accurately presented - explains arguments of considerable complexity - applies problemsolving steps in an organized manner Level 4 80-100% - demonstrates a thorough understanding of concepts - uses the most effective algorithms accurately - tables and graphs accurately and creatively presented - routinely explains complex arguments clearly - applies problemsolving steps in an organized, efficient and clear manner - regular use of - consistent use of Written mathematical mathematical Communication language language - justifies - fully justifies conclusions in an conclusions effective manner effectively - able to apply - able to routinely Application concepts, apply concepts, procedures to procedures to problems seen in problems within a familiar context familiar and some unfamiliar contexts Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity. Page vii Principles of Mathematics - Academic Appendix D (Continued) Written Reports Rubric (Learning Skills) Category Level 1 Level 2 Level 3 Level 4 50-59% 60-69% 70-79% 80-100% - selects resources - selects resources - selects resources - selects resources Works with direction and with assistance effectively independently Independently assistance - uses a variety of - uses a variety of - uses a variety of - uses a variety of learning strategies learning strategies learning strategies learning strategies with direction and with assistance effectively independently assistance - developing - admissible - usually makes a - consistently Organization ability to make a ability to make a plan and can makes a plan and plan and needs plan and needs move to follow independently strategies to supervision to through follows through follow through follow through - rarely meets - occasionally - meets most - meets all Initiative deadlines meets deadlines deadlines deadlines Note: A student whose achievement is below level 1 (50%) has not met the expectations for this assignment or activity. Page viii Principles of Mathematics - Academic
