Mathematics Advanced Mathematics 2200 Interim Edition Curriculum Guide 2012 CONTENTS Contents Acknowledgements ................................................................................................... iii Introduction .....................................................................................................................1 Background ..............................................................................................................................1 Beliefs About Students and Mathematics................................................................................1 Affective Domain......................................................................................................................2 Goals For Students...................................................................................................................2 Conceptual Framework for 10-12 Mathematics ...................................3 Mathematical Processes..........................................................................................................3 Nature of Mathematics.............................................................................................................7 Essential Graduation Learnings..............................................................................................10 Outcomes and Achievement Indicators.................................................................................. 11 Program Organization.............................................................................................................12 Summary ...............................................................................................................................12 Assessment and Evaluation ................................................................................13 Assessment Strategies .........................................................................................................15 Instructional Focus Planning for Instruction . .......................................................................................................17 Teaching Sequence . .............................................................................................................17 Instruction Time Per Unit .......................................................................................................17 Resources . ...........................................................................................................................17 General and Specific Outcomes ....................................................................18 Outcomes with Achievement Indicators Unit 1: Sequences and Series ...............................................................................................19 Unit 2: Trigonometry...............................................................................................................51 Unit 3: Quadratic Functions....................................................................................................67 Unit 4: Quadratic Equations....................................................................................................99 Unit 5: Radical Expressions and Equations . ........................................................................ 119 Unit 6: Rational Expressions and Equations.........................................................................151 Unit 7: Absolute Value and Reciprocal Functions..................................................................181 Unit 8: Systems of Equations............................................................................................... 211 Unit 9: Linear and Quadratic Inequalities..............................................................................229 References .......................................................................................................................251 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 i CONTENTS ii ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ACKNOWLEDGEMENTS Acknowledgements The Department of Education would like to thank the Western and Northern Canadian Protocol (WNCP) for Collaboration in Education, The Common Curriculum Framework for K-9 Mathematics - May 2006 and the Common Curriculum Framework for Grades 10-12 - January 2008, reproduced and/or adapted by permission. All rights reserved. We would also like to thank the provincial Grade 11 Advanced Mathematics curriculum committee and the following people for their contribution: Deanne Lynch, Program Development Specialist - Mathematics Division of Program Development, Department of Education Joanne Hogan, Program Development Specialist - Mathematics Division of Program Development, Department of Education John-Kevin Flynn, Test Development Specialist - Mathematics/Science, Division of Evaluation and Research, Department of Education Mary Ellen Giles, Teacher - Mealy Mountain Collegiate, Happy Valley-Goose Bay Dean Holloway, Teacher - Heritage Collegiate, Lethbridge Kelly House, Teacher - Holy Heart of Mary High, St.John’s Brent Hughes, Teacher - Exploits Valley High, Grand Falls-Windsor Jennifer Lawlor, Teacher - Baltimore School, Ferryland Jim Penney, Teacher - Stephenville High School, Stephenville Clarence White, Teacher - Elwood Regional High, Deer Lake Every effort has been made to acknowledge all sources that contributed to the development of this document. Any omissions or errors will be amended in future printings. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 iii ACKNOWLEDGEMENTS iv ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 INTRODUCTION INTRODUCTION Background The curriculum guide communicates high expectations for students. Beliefs About Students and Mathematics Mathematical understanding is fostered when students build on their own experiences and prior knowledge. The Mathematics curriculum guides for Newfoundland and Labrador have been derived from The Common Curriculum Framework for 1012 Mathematics: Western and Northern Canadian Protocol, January 2008. These guides incorporate the conceptual framework for Grades 10 to 12 Mathematics and the general outcomes, specific outcomes and achievement indicators established in the common curriculum framework. They also include suggestions for teaching and learning, suggested assessment strategies, and an identification of the associated resource match between the curriculum and authorized, as well as recommended, resource materials. Students are curious, active learners with individual interests, abilities and needs. They come to classrooms with varying knowledge, life experiences and backgrounds. A key component in developing mathematical literacy is making connections to these backgrounds and experiences. Students learn by attaching meaning to what they do, and they need to construct their own meaning of mathematics. This meaning is best developed when learners encounter mathematical experiences that proceed from the simple to the complex and from the concrete to the abstract. Through the use of manipulatives and a variety of pedagogical approaches, teachers can address the diverse learning styles, cultural backgrounds and developmental stages of students, and enhance within them the formation of sound, transferable mathematical understandings. Students at all levels benefit from working with a variety of materials, tools and contexts when constructing meaning about new mathematical ideas. Meaningful student discussions provide essential links among concrete, pictorial and symbolic representations of mathematical concepts. The learning environment should value and respect the diversity of students’ experiences and ways of thinking, so that students feel comfortable taking intellectual risks, asking questions and posing conjectures. Students need to explore problem-solving situations in order to develop personal strategies and become mathematically literate. They must come to understand that it is acceptable to solve problems in a variety of ways and that a variety of solutions may be acceptable. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 1 INTRODUCTION Program Design and Components Affective Domain To experience success, students must learn to set achievable goals and assess themselves as they work toward these goals. A positive attitude is an important aspect of the affective domain and has a profound impact on learning. Environments that create a sense of belonging, encourage risk taking and provide opportunities for success help develop and maintain positive attitudes and self-confidence within students. Students with positive attitudes toward learning mathematics are likely to be motivated and prepared to learn, participate willingly in classroom activities, persist in challenging situations and engage in reflective practices. Teachers, students and parents need to recognize the relationship between the affective and cognitive domains, and attempt to nurture those aspects of the affective domain that contribute to positive attitudes. To experience success, students must learn to set achievable goals and assess themselves as they work toward these goals. Striving toward success and becoming autonomous and responsible learners are ongoing, reflective processes that involve revisiting, asssessing and revising personal goals. Goals For Students The main goals of mathematics education are to prepare students to: • use mathematics confidently to solve problems • communicate and reason mathematically Mathematics education must prepare students to use mathematics confidently to solve problems. • appreciate and value mathematics • make connections between mathematics and its applications • commit themselves to lifelong learning • become mathematically literate adults, using mathematics to contribute to society. Students who have met these goals will: • gain understanding and appreciation of the contributions of mathematics as a science, philosophy and art • exhibit a positive attitude toward mathematics • engage and persevere in mathematical tasks and projects • contribute to mathematical discussions • take risks in performing mathematical tasks • exhibit curiosity. 2 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 MATHEMATICAL PROCESSES CONCEPTUAL FRAMEWORK FOR 10-12 MATHEMATICS The chart below provides an overview of how mathematical processes and the nature of mathematics influence learning outcomes. Mathematical Processes There are critical components that students must encounter in a mathematics program in order to achieve the goals of mathematics education and embrace lifelong learning in mathematics. Students are expected to: • communicate in order to learn and express their understanding • Communication [C] • Connections [CN] • connect mathematical ideas to other concepts in mathematics, to everyday experiences and to other disciplines • Mental Mathematics and Estimation [ME] • demonstrate fluency with mental mathematics and estimation • Problem Solving [PS] • develop and apply new mathematical knowledge through problem solving • Reasoning [R] • develop mathematical reasoning • Technology [T] • select and use technologies as tools for learning and for solving problems • Visualization [V] • develop visualization skills to assist in processing information, making connections and solving problems. This curriculum guide incorporates these seven interrelated mathematical processes that are intended to permeate teaching and learning. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 3 MATHEMATICAL PROCESSES Communication [C] Students need opportunities to read about, represent, view, write about, listen to and discuss mathematical ideas. These opportunities allow students to create links between their own language and ideas, and the formal language and symbols of mathematics. Students must be able to communicate mathematical ideas in a variety of ways and contexts. Communication is important in clarifying, reinforcing and modifying ideas, attitudes and beliefs about mathematics. Students should be encouraged to use a variety of forms of communication while learning mathematics. Students also need to communicate their learning using mathematical terminology. Communication helps students make connections among concrete, pictorial, symbolic, oral, written and mental representations of mathematical ideas. Connections [CN] Through connections, students begin to view mathematics as useful and relevant. Contextualization and making connections to the experiences of learners are powerful processes in developing mathematical understanding. When mathematical ideas are connected to each other or to real-world phenomena, students begin to view mathematics as useful, relevant and integrated. Learning mathematics within contexts and making connections relevant to learners can validate past experiences and increase student willingness to participate and be actively engaged. The brain is constantly looking for and making connections. “Because the learner is constantly searching for connections on many levels, educators need to orchestrate the experiences from which learners extract understanding … Brain research establishes and confirms that multiple complex and concrete experiences are essential for meaningful learning and teaching” (Caine and Caine, 1991, p.5). 4 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 MATHEMATICAL PROCESSES Mental Mathematics and Estimation [ME] Mental mathematics and estimation are fundamental components of number sense. Mental mathematics is a combination of cognitive strategies that enhance flexible thinking and number sense. It is calculating mentally without the use of external memory aids. Mental mathematics enables students to determine answers without paper and pencil. It improves computational fluency by developing efficiency, accuracy and flexibility. “Even more important than performing computational procedures or using calculators is the greater facility that students need—more than ever before—with estimation and mental math” (National Council of Teachers of Mathematics, May 2005). Students proficient with mental mathematics “... become liberated from calculator dependence, build confidence in doing mathematics, become more flexible thinkers and are more able to use multiple approaches to problem solving” (Rubenstein, 2001, p. 442). Mental mathematics “... provides the cornerstone for all estimation processes, offering a variety of alternative algorithms and nonstandard techniques for finding answers” (Hope, 1988, p. v). Estimation is used for determining approximate values or quantities or for determining the reasonableness of calculated values. It often uses benchmarks or referents. Students need to know when to estimate, how to estimate and what strategy to use. Estimation assists individuals in making mathematical judgements and in developing useful, efficient strategies for dealing with situations in daily life. Problem Solving [PS] Learning through problem solving should be the focus of mathematics at all grade levels. Learning through problem solving should be the focus of mathematics at all grade levels. When students encounter new situations and respond to questions of the type, “How would you know?” or “How could you ...?”, the problem-solving approach is being modelled. Students develop their own problem-solving strategies by listening to, discussing and trying different strategies. A problem-solving activity requires students to determine a way to get from what is known to what is unknown. If students have already been given steps to solve the problem, it is not a problem, but practice. A true problem requires students to use prior learning in new ways and contexts. Problem solving requires and builds depth of conceptual understanding and student engagement. Problem solving is a powerful teaching tool that fosters multiple, creative and innovative solutions. Creating an environment where students openly seek and engage in a variety of strategies for solving problems empowers students to explore alternatives and develops confident, cognitive mathematical risk takers. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 5 MATHEMATICAL PROCESSES Reasoning [R] Mathematical reasoning helps students think logically and make sense of mathematics. Mathematical reasoning helps students think logically and make sense of mathematics. Students need to develop confidence in their abilities to reason and justify their mathematical thinking. High-order questions challenge students to think and develop a sense of wonder about mathematics. Mathematical experiences in and out of the classroom provide opportunities for students to develop their ability to reason. Students can explore and record results, analyze observations, make and test generalizations from patterns, and reach new conclusions by building upon what is already known or assumed to be true. Reasoning skills allow students to use a logical process to analyze a problem, reach a conclusion and justify or defend that conclusion. Technology [T] Technology contributes to the learning of a wide range of mathematical outcomes and enables students to explore and create patterns, examine relationships, test conjectures and solve problems. Technology contributes to the learning of a wide range of mathematical outcomes and enables students to explore and create patterns, examine relationships, test conjectures and solve problems. Technology can be used to: • explore and demonstrate mathematical relationships and patterns • organize and display data • extrapolate and interpolate • assist with calculation procedures as part of solving problems • decrease the time spent on computations when other mathematical learning is the focus • reinforce the learning of basic facts • develop personal procedures for mathematical operations • create geometric patterns • simulate situations • develop number sense. Technology contributes to a learning environment in which the growing curiosity of students can lead to rich mathematical discoveries at all grade levels. 6 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 NATURE OF MATHEMATICS Visualization [V] Visualization “involves thinking in pictures and images, and the ability to perceive, transform and recreate different aspects of the visual-spatial world” (Armstrong, 1993, p. 10). The use of visualization in the study of mathematics provides students with opportunities to understand mathematical concepts and make connections among them. Visualization is fostered through the use of concrete materials, technology and a variety of visual representations. Visual images and visual reasoning are important components of number, spatial and measurement sense. Number visualization occurs when students create mental representations of numbers. Being able to create, interpret and describe a visual representation is part of spatial sense and spatial reasoning. Spatial visualization and reasoning enable students to describe the relationships among and between 3-D objects and 2-D shapes. Measurement visualization goes beyond the acquisition of specific measurement skills. Measurement sense includes the ability to determine when to measure, when to estimate and which estimation strategies to use (Shaw and Cliatt, 1989). Nature of Mathematics • Change • Constancy Mathematics is one way of trying to understand, interpret and describe our world. There are a number of components that define the nature of mathematics and these are woven throughout this curiculum guide. The components are change, constancy, number sense, patterns, relationships, spatial sense and uncertainty. • Number Sense • Patterns • Relationships • Spatial Sense • Uncertainty Change It is important for students to understand that mathematics is dynamic and not static. As a result, recognizing change is a key component in understanding and developing mathematics. Change is an integral part of mathematics and the learning of mathematics. Within mathematics, students encounter conditions of change and are required to search for explanations of that change. To make predictions, students need to describe and quantify their observations, look for patterns, and describe those quantities that remain fixed and those that change. For example, the sequence 4, 6, 8, 10, 12, … can be described as: • the number of a specific colour of beads in each row of a beaded design • skip counting by 2s, starting from 4 • an arithmetic sequence, with first term 4 and a common difference of 2 • a linear function with a discrete domain (Steen, 1990, p. 184). ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 7 NATURE OF MATHEMATICS Constancy Constancy is described by the terms stability, conservation, equilibrium, steady state and symmetry. Different aspects of constancy are described by the terms stability, conservation, equilibrium, steady state and symmetry (AAASBenchmarks, 1993, p.270). Many important properties in mathematics and science relate to properties that do not change when outside conditions change. Examples of constancy include the following: • The ratio of the circumference of a teepee to its diameter is the same regardless of the length of the teepee poles. • The sum of the interior angles of any triangle is 180°. • The theoretical probability of flipping a coin and getting heads is 0.5. Some problems in mathematics require students to focus on properties that remain constant. The recognition of constancy enables students to solve problems involving constant rates of change, lines with constant slope, direct variation situations or the angle sums of polygons. Number Sense An intuition about number is the most important foundation of a numerate child. Number sense, which can be thought of as intuition about numbers, is the most important foundation of numeracy (British Columbia Ministry of Education, 2000, p.146). A true sense of number goes well beyond the skills of simply counting, memorizing facts and the situational rote use of algorithms. Mastery of number facts is expected to be attained by students as they develop their number sense. This mastery allows for facility with more complex computations but should not be attained at the expense of an understanding of number. Number sense develops when students connect numbers to their own real-life experiences and when students use benchmarks and referents. This results in students who are computationally fluent and flexible with numbers and who have intuition about numbers. The evolving number sense typically comes as a by product of learning rather than through direct instruction. It can be developed by providing rich mathematical tasks that allow students to make connections to their own expereinces and their previous learning. 8 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 NATURE OF MATHEMATICS Patterns Mathematics is about recognizing, describing and working with numerical and non-numerical patterns. Mathematics is about recognizing, describing and working with numerical and non-numerical patterns. Patterns exist in all strands of mathematics. Working with patterns enables students to make connections within and beyond mathematics. These skills contribute to students’ interaction with, and understanding of, their environment. Patterns may be represented in concrete, visual or symbolic form. Students should develop fluency in moving from one representation to another. Students must learn to recognize, extend, create and use mathematical patterns. Patterns allow students to make predictions and justify their reasoning when solving routine and non-routine problems. Learning to work with patterns in the early grades helps students develop algebraic thinking, which is foundational for working with more abstract mathematics. Relationships Mathematics is used to describe and explain relationships. Spatial Sense Spatial sense offers a way to interpret and reflect on the physical environment. Mathematics is one way to describe interconnectedness in a holistic worldview. Mathematics is used to describe and explain relationships. As part of the study of mathematics, students look for relationships among numbers, sets, shapes, objects and concepts. The search for possible relationships involves collecting and analyzing data and describing relationships visually, symbolically, orally or in written form. Spatial sense involves visualization, mental imagery and spatial reasoning. These skills are central to the understanding of mathematics. Spatial sense is developed through a variety of experiences and interactions within the environment. The development of spatial sense enables students to solve problems involving 3-D objects and 2-D shapes and to interpret and reflect on the physical environment and its 3-D or 2-D representations. Some problems involve attaching numerals and appropriate units (measurement) to dimensions of shapes and objects. Spatial sense allows students to make predictions about the results of changing these dimensions; e.g., doubling the length of the side of a square increases the area by a factor of four. Ultimately, spatial sense enables students to communicate about shapes and objects and to create their own representations. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 9 ESSENTIAL GRADUATION LEARNINGS Uncertainty Uncertainty is an inherent part of making predictions. In mathematics, interpretations of data and the predictions made from data may lack certainty. Events and experiments generate statistical data that can be used to make predictions. It is important to recognize that these predictions (interpolations and extrapolations) are based upon patterns that have a degree of uncertainty. The quality of the interpretation is directly related to the quality of the data. An awareness of uncertainty allows students to assess the reliability of data and data interpretation. Chance addresses the predictability of the occurrence of an outcome. As students develop their understanding of probability, the language of mathematics becomes more specific and describes the degree of uncertainty more accurately. Essential Graduation Learnings Essential graduation learnings are statements describing the knowledge, skills and attitudes expected of all students who graduate from high school. Essential graduation learnings are cross-curricular in nature and comprise different areas of learning: aesthetic expression, citizenship, communication, personal development, problem solving, technological competence and spiritual and moral development. Aesthetic Expression Graduates will be able to respond with critical awareness to various forms of the arts and be able to express themselves through the arts. Citizenship Graduates will be able to assess social, cultural, economic and environmental interdependence in a local and global context. Communication Graduates will be able to use the listening, viewing, speaking, reading and writing modes of language(s) and mathematical and scientific concepts and symbols to think, learn and communicate effectively. Personal Development Graduates will be able to continue to learn and to pursue an active, healthy lifestyle. Problem Solving Graduates will be able to use the strategies and processes needed to solve a wide variety of problems, including those requiring language and mathematical and scientific concepts. 10 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 OUTCOMES Technological Competence Graduates will be able to use a variety of technologies, demonstrate an understanding of technological applications, and apply appropriate technologies for solving problems. Spiritual and Moral Development Graduates will be able to demonstrate an understanding and appreciation for the place of belief systems in shaping the development of moral values and ethical conduct. See Foundations for the Atlantic Canada Mathematics Curriculum, pages 4-6. The mathematics curriculum is designed to make a significant contribution towards students’ meeting each of the essential graduation learnings (EGLs), with the communication, problem-solving and technological competence EGLs relating particularly well to the mathematical processes. Outcomes and Achievement Indicators The curriculum is stated in terms of general outcomes, specific outcomes and achievement indicators. General Outcomes General outcomes are overarching statements about what students are expected to learn in each course. Specific Outcomes Specific outcomes are statements that identify the specific skills, understanding and knowledge that students are required to attain by the end of a given course. In the specific outcomes, the word including indicates that any ensuing items must be addressed to fully meet the learning outcome. The phrase such as indicates that the ensuing items are provided for illustrative purposes or clarification, and are not requirements that must be addressed to fully meet the learning outcome. Achievement Indicators Achievement indicators are samples of how students may demonstrate their achievement of the goals of a specific outcome. The range of samples provided is meant to reflect the scope of the specific outcome. Specific curriculum outcomes represent the means by which students work toward accomplishing the general curriculum outcomes and ultimately, the essential graduation learnings. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 11 PROGRAM ORGANIZATION Program Organization Program Level Course 1 Course 2 Course 3 Course 4 Advanced Academic Applied Mathematics 1201 Mathematics 1202 Mathematics 2200 Mathematics 2201 Mathematics 2202 Mathematics 3200 Mathematics 3201 Mathematics 3202 Mathematics 3208 The applied program is designed to provide students with the mathematical understandings and critical-thinking skills identified for entry into the majority of trades and for direct entry into the workforce. The academic and advanced programs are designed to provide students with the mathematical understandings and critical-thinking skills identified for entry into post-secondary programs. Students who complete the advanced program will be better prepared for programs that require the study of calculus. The programs aim to prepare students to make connections between mathematics and its applications and to become numerate adults, using mathematics to contribute to society. Summary 12 The conceptual framework for Grades 10-12 Mathematics (p. 3) describes the nature of mathematics, mathematical processes and the mathematical concepts to be addressed. The components are not meant to stand alone. Activities that take place in the mathematics classroom should result from a problem-solving approach, be based on mathematical processes and lead students to an understanding of the nature of mathematics through specific knowledge, skills and attitudes among and between topics. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ASSESSMENT ASSESSMENT AND EVALUATION Purposes of Assessment What learning is assessed and evaluated, how it is assessed and evaluated, and how results are communicated send clear messages to students and others about what is really valued. Assessment techniques are used to gather information for evaluation. Information gathered through assessment helps teachers determine students’ strengths and needs in their achievement of mathematics and guides future instructional approaches. Teachers are encouraged to be flexible in assessing the learning success of all students and to seek diverse ways in which students might demonstrate what they know and are able to do. Evaluation involves the weighing of the assessment information against a standard in order to make an evaluation or judgment about student achievement. Assessment has three interrelated purposes: • assessment for learning to guide and inform instruction; • assessment as learning to involve students in self-assessment and setting goals for their own learning; and • assessment of learning to make judgements about student performance in relation to curriculum outcomes. Assessment for Learning Assessment for learning involves frequent, interactive assessments designed to make student understanding visible. This enables teachers to identify learning needs and adjust teaching accordingly. It is an ongoing process of teaching and learning. Assessment for learning: • requires the collection of data from a range of assessments as investigative tools to find out as mush as possible about what students know • provides descriptive, specific and instructive feedback to students and parents regarding the next stage of learning • actively engages students in their own learning as they assess themselves and understand how to improve performance. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 13 ASSESSMENT Assessment as Learning Assessment as learning actively involves students’ reflection on their learning and monitoring of their own progress. It focuses on the role of the student as the critical connector between assessment and learning, thereby developing and supporting metacognition in students. Assessment as learning: • supports students in critically analysing their learning related to learning outcomes • prompts students to consider how they can continue to improve their learning • enables students to use information gathered to make adaptations to their learning processes and to develop new understandings. Assessment of Learning Assessment of learning involves strategies to confirm what students know, demonstrate whether or not they have met curriculum outcomes, or to certify proficiency and make decisions about students’ future learning needs. Assessment of learning occurs at the end of a learning experience that contributes directly to reported results. Traditionally, teachers relied on this type of assessment to make judgments about student performance by measuring learning after the fact and then reporting it to others. Used in conjunction with the other assessment processes previously outlined, however, assessment of learning is strengthened. Assessment of learning: • provides opportunities to report evidence to date of student achievement in relation to learning outcomes, to parents/guardians and other stakeholders • confirms what students know and can do • occurs at the end of a learning experience using a variety of tools. Because the consequences of assessment of learning are often farreaching, teachers have the responsibility of reporting student learning accurately and fairly, based on evidence obtained from a variety of contexts and applications. 14 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ASSESSMENT Assessment Strategies Assessment techniques should match the style of learning and instruction employed. Several options are suggested in this curriculum guide from which teachers may choose, depending on the curriculum outcomes, the class and school/district policies. Observation (formal or informal) This technique provides a way of gathering information fairly quickly while a lesson is in progress. When used formally, the student(s) would be aware of the observation and the criteria being assessed. Informally, it could be a frequent, but brief, check on a given criterion. Observation may offer information about the participation level of a student for a given task, use of a concrete model or application of a given process. The results may be recorded in the form of checklists, rating scales or brief written notes. It is important to plan in order that specific criteria are identified, suitable recording forms are ready, and all students are observed within a reasonable period of time. Performance This curriculum encourages learning through active participation. Many of the curriculum outcomes promote skills and their applications. In order for students to appreciate the importance of skill development, it is important that assessment provide feedback on the various skills. These may be the correct manner in which to use a manipulative, the ability to interpret and follow instructions, or to research, organize and present information. Assessing performance is most often achieved through observing the process. Paper and Pencil These techniques can be formative or summative. Whether as part of learning, or a final statement, students should know the expectations for the exercise and how it will be assessed. Written assignments and tests can be used to assess knowledge, understanding and application of concepts. They are less successful at assessing processes and attitudes. The purpose of the assessment should determine what form of paper and pencil exercise is used. Journal Journals provide an opportunity for students to express thoughts and ideas in a reflective way. By recording feelings, perceptions of success, and responses to new concepts, a student may be helped to identify his or her most effective learning style. Knowing how to learn in an effective way is powerful information. Journal entries also give indicators of developing attitudes to mathematical concepts, processes and skills, and how these may be applied in the context of society. Selfassessment, through a journal, permits a student to consider strengths and weaknesses, attitudes, interests and new ideas. Developing patterns may help in career decisions and choices of further study. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 15 ASSESSMENT Interview This curriculum promotes understanding and applying mathematics concepts. Interviewing a student allows the teacher to confirm that learning has taken place beyond simple factual recall. Discussion allows a student to display an ability to use information and clarify understanding. Interviews may be a brief discussion between teacher and student or they may be more extensive. Such conferences allow students to be proactive in displaying understanding. It is helpful for students to know which criteria will be used to assess formal interviews. This assessment technique provides an opportunity to students whose verbal presentation skills are stronger than their written skills. Presentation The curriculum includes outcomes that require students to analyze and interpret information, to be able to work in teams, and to communicate information. These activities are best displayed and assessed through presentations. These can be given orally, in written/pictorial form, by project summary, or by using electronic systems such as video or computer software. Whatever the level of complexity, or format used, it is important to consider the curriculum outcomes as a guide to assessing the presentation. The outcomes indicate the process, concepts and context for which a presentation is made. Portfolio Portfolios offer another option for assessing student progress in meeting curriculum outcomes over a more extended period of time. This form of assessment allows the student to be central to the process. There are decisions about the portfolio, and its contents, which can be made by the student. What is placed in the portfolio, the criteria for selection, how the portfolio is used, how and where it is stored, and how it is evaluated are some of the questions to consider when planning to collect and display student work in this way. The portfolio should provide a long-term record of growth in learning and skills. This record of growth is important for individual reflection and self-assessment, but it is also important to share with others. For all students, it is exciting to review a portfolio and see the record of development over time. 16 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 INSTRUCTIONAL FOCUS INSTRUCTIONAL FOCUS Planning for Instruction Consider the following when planning for instruction: • Integration of the mathematical processes within each topic is expected. • By decreasing emphasis on rote calculation, drill and practice, and the size of numbers used in paper and pencil calculations, more time is available for concept development. • Problem solving, reasoning and connections are vital to increasing mathematical fluency and must be integrated throughout the program. • There should be a balance among mental mathematics and estimation, paper and pencil exercises, and the use of technology, including calculators and computers. Concepts should be introduced using manipulatives and be developed concretely, pictorially and symbolically. • Students bring a diversity of learning styles and cultural backgrounds to the classroom. They will be at varying developmental stages. Teaching Sequence The curriculum guide for Advanced Mathematics 2200 is organized by units. This is only a suggested teaching order for the course. There are a number of combinations of sequences that would be appropriate. Each two page spread lists the topic, general outcome, and specific outcome. Instruction Time Per Unit The suggested number of hours of instruction per unit is listed in the guide at the beginning of each unit. The number of suggested hours includes time for completing assessment activities, reviewing and evaluating. The timelines at the beginning of each unit are provided to assist in planning. The use of these timelines is not mandatory. However, it is mandatory that all outcomes are taught during the school year, so a long term plan is advised. Teaching of the outcomes is ongoing, and may be revisited as necessary. Resources The authorized resource for Newfoundland and Labrador students and teachers is Pre-Calculus 11 (McGraw-Hill Ryerson). Column four of the curriculum guide references Pre-Calculus 11 for this reason. Teachers may use any other resource, or combination of resources, to meet the required specific outcomes. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 17 GENERAL AND SPECIFIC OUTCOMES GENERAL AND SPECIFIC OUTCOMES GENERAL AND SPECIFIC OUTCOMES WITH ACHIEVEMENT INDICATORS (pages 19-214) This section presents general and specific outcomes with corresponding achievement indicators and is organized by unit. The list of indicators contained in this section is not intended to be exhaustive but rather to provide teachers with examples of evidence of understanding that may be used to determine whether or not students have achieved a given specific outcome. Teachers may use any number of these indicators or choose to use other indicators as evidence that the desired learning has been achieved. Achievement indicators should also help teachers form a clear picture of the intent and scope of each specific outcome. Advanced Mathematics 2200 is organized into nine units: Sequences and Series, Trigonometry, Quadratic Functions, Quadratic Equations, Radical Expressions and Equations, Rational Expressions and Equations, Absolute Value and Reciprocal Functions, Systems of Equations and Linear and Quadratic Inequalities. 18 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Sequences and Series Suggested Time: 13 Hours SEQUENCES AND SERIES Unit Overview Focus and Context Outcomes Framework In this unit, students will differentiate between finite and infinite sequences. They will then explore and develop a formula for the general term of arithmetic and geometric sequences, and progress to arithmetic and geometric series. Students will determine if a given series is convergent or divergent. They will then write the general equation for the sum of an infinite geometric series. The concepts of sequences and series will be applied to solve problems. GCO Develop algebraic and graphical reasoning through the study of relations. SCO RF9 Analyze arithmetic sequences and series to solve problems. SCO RF10 Analyze geometric sequences and series to solve problems. 20 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Mathematics 2200 Mathematics 3200 Relations and Functions RF9 Analyze arithmetic sequences and series to solve problems. [CN, PS, R] RF10 Analyze geometric sequences and series to solve problems. [PS, R] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 21 SEQUENCES AND SERIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF9 Analyze arithmetic sequences and series to solve problems. [CN, PS, R] Achievement Indicators: RF9.1 Identify the assumption(s) made when defining an arithmetic sequence or series. RF9.2 Provide and justify an example of an arithmetic sequence. In Grade 9, students were exposed to linear patterns (9PR1). In Mathematics 1201, they determined the equation of a linear relation (RF7). In this unit, students will explore the concept of arithmetic sequences and series. They will work with patterns where consecutive terms produce a common difference. Students will write a formula for the general term of an arithmetic sequence and solve for any missing values. Once sequences have been explored, students will be introduced to an arithmetic series. They will make the connection that an arithmetic series is the sum of an arithmetic sequence, derive a formula and apply it in a variety of problem solving situations. Students should investigate the concept of a sequence by observing a variety of number patterns to develop an understanding of the notation, symbols and domain associated with arithmetic sequences (i.e. terms t1, t2, finite sequence, infinite sequence, common difference d, and the general term tn). In Mathematics 1201, students were exposed to the concept of domain as it relates to linear functions (RF1). Provide students with an arithmetic sequence and ask them to graph the terms of the sequence. Students should make the connection that the terms of the sequence are the ordered pairs of the graph. Ask them what the x and y-values of the ordered pairs represent and what number system they belong to. They should recognize the domain of the sequence is the set of natural numbers. Encourage students to provide their own examples of arithmetic sequences and justify, finding the difference between consecutive terms, why such a sequence is arithmetic. They should not assume every sequence that has an increasing or decreasing pattern in addition is arithmetic. Students should conclude, however, that if a sequence is arithmetic the same number is being added to each term in the sequence. RF9.3 Derive a rule for determining the general term of an arithmetic sequence. Use a numerical example to give students an opportunity to identify patterns and a possible rule for an arithmetic sequence. Consider a sequence such as 5, 7, 9, 11, 13,...... and the following questions to guide student discussion. • What is the common difference? • Can this sequence be rewritten to show the pattern of the first term and the common difference? • Can you predict the formula for the general term of an arithmetic sequence based on the pattern 5, 5 + 2, 5 + 2(2), 5 + 3(2),.....? • 22 Can you write the pattern in general terms for any first term and common difference? ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to determine the first five terms for each of the sequences. (i) tn = 2n - 1 2 (ii) t n = 4 n − 4 n + 1 2n 2 +n −1 (iii) t n = n +1 Ask them if each sequence is arithmetic since the first five terms result in {1, 3, 5, 7, 9}. They should explain their reasoning. (RF9.2) Observation • Ask students to explore the following sequences to determine whether each is arithmetic. They should justify their conclusions. 1.1 Arithmetic Sequences Student Book (SB): pp.6-21 Teacher Resource (TR): pp.10-18 Blackline Master (BLM): 1-3, 1-4 (i) 5.3, 5.9, 6.5 (ii) x - 1, x + 1, x + 3 (iii) x1, x2, x3, (iv) 2x + 5, 4x + 5, 6x + 5 (RF9.2) Journal • Ask students to explain the term “sequence” in their own words. Ask them to compare this to the mathematical definition found at www.en.wikipedia.org/wiki/Sequence. They should explain how the two definitions are similar and how they are different. (RF9.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 23 SEQUENCES AND SERIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF9 Continued ... Achievement Indicators: RF9.2, RF9.3 Continued Students should notice that the terms of the arithmetic sequence with first term t1 and a common difference d can be written as t1 = t1, t2 = t1 + d, t3 = t1 + 2d, t4 = t1 + 3d... Using this pattern, they can write the formula for any arithmetic sequence tn = t1 + (n - 1)d. RF9.4 Determine t1 , d, n, or tn in a problem that involves an arithmetic sequence. Give students an opportunity to write their own examples of arithmetic sequences and to write the appropriate formula. They should use numerical and algebraic examples. Encourage them to list t1, d, and n, and then write the equation for tn. RF9.5 Solve a problem that involves an arithmetic sequence or series. When solving problems involving arithmetic sequences consider a variety of examples such as, but not limited to, the following: • Find the general term tn using tables, charts, graphs or an equation. • Find the number of terms in a finite arithmetic sequence when given the value of the nth term. • Find the common difference when provided with algebraic expressions representing the value of terms for the sequence. When provided with the general term, students sometimes have difficulty differentiating between the term and the term number. When they are asked to find the tenth term, for example, they may be unsure whether to write t10 or tn = 10. It is important to reinforce that t10 represents the tenth term while tn = 10 represents the nth term having a value of 10. RF9.6 Describe the relationship between arithmetic sequences and linear functions. 24 Students will make the connection that arithmetic sequences, when graphed, form linear graphs. Students may need to review graphing linear functions using a table of values and/or the slope y-intercept method. They should observe the following features when comparing an arithmetic sequence to a linear function: • The slope is the common difference. • The y-intercept is the initial value minus the common difference. • The domain of a sequence is the set of natural numbers, while the domain of a linear function, based on the context, can be the set of real numbers. • The graph of an arithmetic sequence consists of discrete data, while the graph of a linear function may be discrete or continuous. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • • Ask students to create two arithmetic sequences and find the general term tn. Ask them to determine the sum and then the product of the two sequences. They should explain if these new sequences created are arithmetic and justify their reasoning. (RF9.2, RF9.4) Pre-Calculus 11 1.1 Arithmetic Sequences SB: pp.6-21 TR: pp.10-18 BLM: 1-3, 1-4 For each arithmetic sequence, ask students to write a formula for tn and use it to find the indicated term. (i) -4, 1, 6, 11, … , t13 (ii) 9, 1, -7, -15, …, t46 (RF9.4, RF9.5) • Consecutive terms of an arithmetic sequence are (5 + x), 8, and (1 + 2x). Ask students to determine the value of x. (RF9.4, RF9.5) • The diagram shows a pattern of positive integers in five columns. If the pattern is continued, ask students to determine the columns in which these numbers will appear. (iii) 301 Col 1 Col 2 Col 3 Col 4 Col 5 1 2 3 5 4 (iv) 8725 6 (i) 49 (ii) 117 7 10 11 8 9 12 15 13 14 (RF9.4, RF9.5) Interview • Ask students to consider the following problem and explain if the calculation is correct. They should justify their reasoning. Paul must determine the 50th term of an arithmetic sequence beginning with 5 and having a common difference of 9. He calculates (50 x 9) + 5 = 455. (RF9.4, RF9.5) • Zachary is having trouble remembering the formula tn = t1 + (n - 1)d correctly. He thinks the formula should be tn = t1 + nd. Ask students how they would explain to Zachary that he should use (n - 1)d rather than nd in the formula. (RF9.4, RF9.5) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 25 SEQUENCES AND SERIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF9 Continued ... Achievement Indicators: The relationship bewtween an arithmetic sequence and a linear function can be highlighted through a discussion such as the following: n tn RF9.6 Continued 1 3 2 5 3 7 4 9 5 11 • Find the general term tn . Compare this equation to a linear equation y = mx + b. Are they similar? • Graph the sequence. What do you notice? • What is the relationship between the slope of the line and the common difference? • Evaluate the first term minus the common difference of the sequence. What does this value represent in the linear equation? The relationship between arithmetic sequences and linear functions could also be developed using a symbolic approach. RF9.7 Derive a rule for determining the sum of n terms of an arithmetic series. tn = t1 + (n - 1)d tn = t1 + dn - d tn = dn + t1 - d tn = dn + (t1 - d) vs y = mx + b where d = m and t1 - d = b Students will write the sum of the first n terms for any arithmetic sequence. They should work with a numerical example and based on their observations, write a formula for the general case. The following example provides an opportunity to review arithmetic sequences and introduce arithmetic series. • Suppose you create a Facebook account and add one new friend on day 1, four more friends on day 2 and four more friends each day for a total of 25 days. How many friends would you have in total after the 25th day? Use the following questions to guide students: • What is the arithmetic sequence? • What is the first term? • What is the rule for tn? What is the last term? • What do the sum of the terms in the sequence represent? • How many terms are we summing up? • How can the value of S25 be determined? Although some students may just add up all the terms in the sequence, they should realize this is not the most efficient method. 26 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 1.1 Arithmetic Sequences Ask students to complete the following activity using technology: n tn 1 1 2 -1 3 -3 4 -5 5 -7 SB: pp.6-21 TR: pp.10-18 BLM: 1-3, 1-4 (i) Find a formula for the general term tn and record your findings. (ii) Graph the discrete data. (iii) Draw a line through the points and record the equation. (iv) Compare your results from steps (i) and (iii). How does the common difference relate to the slope? (v) Subtract d from the first term of the sequence. Compare this value to the y-intercept from the line of best fit. What do you notice? (RF9.6) 1.2 Arithmetic Series SB: pp.22-31 TR: pp.19-25 BLM: 1-3, 1-5 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 27 SEQUENCES AND SERIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF9 Continued ... Achievement Indicator: RF9.7 Continued Introduce students to Gauss’s method of determining the sum of the first 100 positive integers. Use this method to determine S25 in the previous example. Ask students to write the series twice, once in ascending order and the other in descending order. Then, sum the two series. S25 = 1 + 5 + 9 +.......... + 89 + 93 + 97 +S25 = 97 + 93 + 89 +.......... + 9 + 5 + 1 2S25 = 98 + 98 + 98 +.......... + 98 + 98 + 98 2S25 = 25(98) 25 S25 = (98) 2 Encourage students to try to determine where the different parts of that formula come from. When they are trying to make a conjecture about the general case, it is best to leave the original values rather than simplifying the expression. Ask students what they think the values 25 and 98 represent. They should recognize that 25 is the number of terms and 98 is the sum of 1 and 97, representing t1 and t25. Therefore, S25 = 25 2 ( t 1 + t 25 ) . The algebraic approach should also be explored to derive the formula for the sum of the general arithmetic series, Sn = n2 ( t 1 + t n ). The sum of n terms of an arithmetic series is represented by Sn = t1 + t1 + d + .... t1 + (n - 2)d + t1 + (n - 1)d . Using Gauss’s method, guide students through the following derivation: Sn = + t1 Sn = t 1 + ( n − 1)d t1 + d + ..... t 1 + ( n − 2)d + t 1 + ( n − 2)d + ..... 2Sn = 2t 1 + ( n − 1)d + 2t 1 + ( n − 1)d + ..... 2 Sn t1 + d 2t 1 + ( n − 1)d + t 1 + ( n − 1)d + t1 + 2t 1 + ( n − 1)d = n(2t 1 + ( n − 1)d ) 2Sn = n( t 1 + t 1 + ( n − 1)d ) tn 2Sn = n( t 1 + t n ) Sn = 28 n ( t 1 +t n ) 2 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • Ask students to use Gauss’s method to determine the sum of the first 12 positive integers. (RF9.7) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Pre-Calculus 11 1.2 Arithmetic Series SB: pp.22-31 TR: pp.19-25 BLM: 1-3, 1-5 29 SEQUENCES AND SERIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF9 Continued ... Achievement Indicators: RF9.8 Determine t1, d, n, or Sn in a problem that involves an arithmetic series. RF9.5 Continued Students will evaluate and/or manipulate the given arithmetic series to determine the first term, the common difference, the number of terms in the sequence, the nth term or the sum. They should solve problems that involve an arithmetic series within a context. Consider the following example: • An auditorium has 20 seats on the first row, 24 seats on the second row, 28 seats on the third row, and so on, and has 30 rows of seats. How many seats are in the theatre? To solve this problem, students need to ask themselves the following questions: 30 • What is the problem asking? • What is the pattern? Does it represent an arithmetic sequence? • What information do we need to find? ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • A round robin tournament is to be played by 3 teams. Ask students to determine the total number of games required to guarantee that each team plays the other exactly once. Repeat the same activity for a tournament consisting of 4 teams. Ask students to use their findings from above to find the total number of games for a national round robin tournament with all ten provinces and three territories participating. They should explain how their answer is related to an arithmetic series. Pre-Calculus 11 1.2 Arithmetic Series SB: pp.22-31 TR: pp.19-25 BLM: 1-3, 1-5 (RF9.5, RF9.8) • Ask students to find the sum of 20 + 14 + 8 + … + (-70). (RF9.8) • A theater has 60 seats in the first row, 68 seats in the second row, 76 seats in the third row, and so on in the same increasing pattern. If the theater has 20 rows of seats, how many seats are in the theater? (RF9.5, RF9.8) • In an arithmetic series t1 = 6 and S9 = 108. Ask students to find the common difference and the sum of the first 20 terms. (RF9.8) • For three months in the summer (12 weeks), Job A pays $325 per month with a monthly raise of $100. Job B pays $50 per week with a weekly raise of $10. Ask students which is the better paying job and why. (RF9.5, RF9.8) • Ask students to show that 1 + 3 + 5 + … + (2n - 1) = n2. (RF9.5, RF9.8) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 31 SEQUENCES AND SERIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF10 Analyze geometric sequences and series to solve problems. [PS, R] Students will continue to explore patterns with a focus shifting to geometric sequences and series. They will explore examples of geometric sequences having a “common ratio” pattern resulting in a specific formula. Students will then differentiate between a geometric sequence and a geometric series. Achievement Indicators: RF10.1 Identify assumptions made when identifying a geometric sequence or series. RF10.2 Provide and justify an example of a geometric sequence. RF10.3 Derive a rule for determining the general term of a geometric sequence. RF10.4 Determine t1, r, n or tn in a problem that involves a geometric sequence. Similar to an arithmetic sequence, students will continue to use the same notation when working with the terms of a geometric sequence. The symbol r will represent the common ratio. Students should provide their own examples of geometric sequences and illustrate the concept of a common ratio by determining the quotient t of pairs of consecutive terms (i.e., tn+1 ), demonstrating why any such n sequence is geometric. It is important for students to think about what makes a sequence geometric as opposed to arithmetic. They should differentiate between a common difference and a common ratio. To develop the formula for a geometric sequence, ask students to consider a sequence such as 1, 3, 9, 27, 81,...... Use the following questions to guide them through the process: • What is the common ratio? • Can this sequence be rewitten to show the pattern of the first term and the common ratio? • Can you predict the formula of a geometric sequence based on the pattern 1, 1(3)1, 1(3)2, 1(3)3, 1(3)4.....? • Can you write the pattern in general terms for any first term and common ratio ? Students should notice that the terms of the geometric sequence with first term t1 and a common ratio r can be written in general as t1= t1, t2 = t1(r)1, t3 = t1(r)2, t4 = t1(r)3... They should recognize the formula for an geometric sequence is tn = t1 (rn - 1). Ask students to draw a sketch of the geometric sequence where the x-axis represents the number of the term and the y-axis represents the term. They should notice the terms of the geometric sequence are not linear. Exposure to the graph of an exponential function may be warranted here since the graph of a geometric sequence is an exponential function with a domain belonging to the natural numbers. Students will study exponential functions and their graphs in Mathematics 3200. Provide numerical and algebraic examples for students to evaluate and/ or manipulate a given geometric sequence to determine the first term, the common ratio, the number of terms in the finite sequence, or the general rule. 32 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Provide students with pieces of square paper of varying side lengths, n. Ask students to fold the paper in half repeatedly. After each fold, the area of the top surface is measured and recorded. Ask them what type of sequence is formed and to explain their reasoning. (RF10.2) 1.3 Geometric Sequences SB: pp.32-45 TR: pp.26-33 BLM: 1-3, 1-6 Paper and Pencil • Ask students to determine the first three terms of each of the following sequences: tn = 3n and tn = 3(3n - 1). Ask them what they notice about the value of t1, t2 and t3 and the type of sequences created. They should explain their reasoning. (RF10.1, RF10.2) • The formula for a geometric sequence is given by tn = t1 (r n - 1). Ask students to create examples of various geometric sequences in this form and explain what happens if n = 1. Ask them what type of a sequence they will continually produce if n = 1. (RF10.4) • You are hired to complete a job for a month that offers two different payment options. In plan A, the payment begins with $3 on day 1, $6 on day 2, $9 on day 3, etc. In plan B, the payment begins with $0.01 on day 1, $0.02 on day 2, $0.04 on day 3, etc. Ask students to determine which payment plan would be more feasible. They should explain why an employer might or might not offer a payment such as plan B. (RF10.4) Interview • A classmate is having trouble with the formula tn = t1 (r n - 1). He thinks the formula should be tn = t1( r n) . Ask students how they would explain to him that he should use r n - 1 rather than r n in the formula. (RF10.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 33 SEQUENCES AND SERIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF10 Continued ... Achievement Indicators: RF10.5 Derive a rule for determining the sum of n terms of a geometric series. Students will first work with finite geometric sereis. With teacher support, students will derive a nformula for the sum of the first n terms t ( r −1) for a geometric series, Sn = 1 r −1 . Encourage students to test the formula using the following numerical example. This would give them an opportunity to review geometric sequences while being introduced to geometric series. Consider the example: • A student is constructing a family tree. She is hoping to trace back through 10 generations to calculate the total number of ancestors he has. Determine the total number of ancestors after the 10th generation. Students should recognize every person has 2 parents, 4 grandparents, 8 great-grandparents, and so on. Therefore, the number of ancestors through ten generations is 2 + 4 + 8+ 16 + 32 + 64 + 128 + 256 + 512 + 1024. Let S represent the sum of this series. Ask students to multiply it by the common ratio, 2. S10= 2 + 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 2S10 = 4 + 8 + 16 + 32 + 64 + 128 + 256 + 512 + 1024 + 2048 2S10-1S10 = -2 + 2048 Students calculate the sum of the first ten terms of the series to be 2046. Going back through the generations, each person has 2046 ancestors. RF10.6 Determine t1 , r, n or Sn in a problem that involves a geometric series. RF10.7 Solve a problem that involves a geometric sequence or series. Students will evaluate and/or manipulate the given geometric series to determine the first term, the common ratio, the number of terms in the sequence, the nth term or the finite sum. It is also important to solve contextual problems that involves a geometric series. Consider a ball that is dropped from a height of 38.28 m and bounces back up 60% of the original height. Ask students to find the total distance travelled by the ball by the time it hits the ground for the tenth time. When working through the problems, remind students of the following guidelines: • • • • • • 34 Draw a diagram if necessary. Write out the terms of the sequence. Determine if the problem is a sequence or series. Determine if the problem is arithmetic or geometric. Construct the formula using the given information in the problem. Solve the problem. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Interview • Ask students to respond to the following questions: (i) Explain why there can be no infinite geometric series with a first term of 12 and a sum of 5. (ii) Explain why the sum of an infinite geometric series is positive if and only if the first term is positive. (RF10.7) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Resources/Notes Authorized Resource Pre-Calculus 11 1.4 Geometric Series SB: pp.46-57 TR: pp.34-40 BLM: 1-3, 1-7 35 SEQUENCES AND SERIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF10 Continued ... Achievement Indicators: RF10.8 Explain why a geometric series is convergent or divergent. RF10.9 Generalize, using inductive reasoning, a rule for determining the sum of an infinite geometric series. To find the sum of an infinite geometric series, Sn = 1t−1r where r ≠ 1, students will first need to differentiate between a convergent and divergent series. To develop the idea of convergence or divergence, expose students to a variety of geometric sequences where the common ratio is different for each. Ask students to determine the partial sum by adding the first two terms, the first three terms and so on. They will check to see if the partial sum approaches a particular value as the number of terms get larger. 1 2 Geometric Series Partial Sum 2 + 4 + 8 +16+... (r > 1) S1 = 2, S2 = 6, S3 = 14 + 41 + 81 + 161 + .... S1 = 1 2 S2 = , 3 4 , S3 = 87 Convergent/ Divergent diverges converges to 1 (-1 < r < 1) -1 + 1+ -1 + 1+ .... (r = -1) S1 = -1, S2 = 0, S3 = -1 S4 = 0 diverges 1 + 1 + 1 + 1 +... (r = 1) S1 = 1, S2 = 2, S3 = 3, S4 = 4 diverges Once students determine the common ratio for each geometric series, they should state their conclusions about convergence or divergence based on the value of r. Ask students why the value of r cannot equal 0. Students could use a concrete representation to help with their understanding of an infinite geometric series. Using a string, 1 metre long, students will cut it in half and place one of the halves stretched out on a table. Using the remaining half, they will then cut it in half so that 1 they have two quarters. Place one of the quarters at the end of the 2 string on the table. They now have 43 of the string on the table. Halve the remaining quarter string so they have two eighths and place one of the eighths at the end of the string on the table. They now have 87 of the string on the table, and so on. The goal is for students to recognize that this infinite series represents a sequence of partial sums. Students can 1 2 verify, using the formula Sn = 1− 1 , that the sum should equal 1. 2 36 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SEQUENCES AND SERIES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to determine whether the following sequences converge or diverge. They should explain your reasoning. (i) 8, 4, 2, 1, 0.5, .... 1.5 Infinite Geometric Series SB: pp.58-65 TR: pp.41-46 BLM: 1-3, 1-8 (ii) 3, 73 , 53 ,1, 13 ,..... (iii) 5-3, 5-2, 5-1, 50, ... (iv) t1 + d, t1 + 2d, t1 + 3d, t1 + 4d, ... (RF10.8) • The midpoints of a square with sides 1 m long are joined to form another square. Then the midpoints of the sides of the second square are joined to form a third square. This process is continued indefinitely to form an infinite set of smaller and smaller squares converging on the center of the original square. Ask students to determine the total length of the segments forming the sides of all the squares. (RF10.9) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 37 SEQUENCES AND SERIES 38 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Trigonometry Suggested Time: 14 Hours TRIGONOMETRY Unit Overview Focus and Context In this unit, students will be exposed to angles in standard position where 0 °≤ θ ≤ 360°. They will also be introduced to reference angles and quadrantal angles. Students will determine the exact values of the primary trigonometric ratios for angles between 0° and 360°. Students will use the sine law and the cosine law to determine unknown side lengths and angle measures in oblique triangles. They will be asked to explain their thinking about which law to use. Outcomes Framework GCO Develop trigonometric reasoning. SCO T1 Demonstrate an understanding of angles in standard position [0° to 360°]. SCO T2 Solve problems, using the three primary trigonometric ratios for angles from 0° to 360° in standard position. SCO T3 Solve problems, using the cosine law and sine law, including the ambiguous case. 40 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Measurement M4 Develop and apply the primary trigonometric ratios (sine, cosine, tangent) to solve problems that involve right triangles. [C, CN, PS, R, T, V] Mathematics 2200 Mathematics 3200 Trigonometry Trigonometry T1 Demonstrate an understanding T1 Demonstrate an understanding of angles in standard position [0° of angles in standard position, to 360°]. expressed in degrees and radians. [CN, ME, R, V] [R, V] T2 Develop and apply the equation of the unit circle. T2 Solve problems, using the three [CN, R, V] primary trigonometric ratios for angles from 0° to 360° in standard T3 Solve problems, using the six trigonometric ratios for angles position. expressed in radians and degrees. [C, ME, PS, R, T, V] [ME, PS, R, T, V] T3 Solve problems, using the cosine law and sine law, including the ambigous case. [C, CN, PS, R, T] T4 Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems. [CN, PS, T, V] T5 Solve, algebraically and graphically, first and seond degree trigonometric equations with the domain expressed in degrees and radians. [CN, PS, R, T, V] T6 Prove trigonometric identities, using: • reciprocal identities • quotient identities • Pythagorean identities • sum or difference identities (restricted to sine, cosine and tangent) • double-angle identities (restricted to sine, cosine and tangent) [R, T, V] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 41 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T1 Demonstrate an understanding of angles in standard position [0° to 360°]. [R, V] In Mathematics 1201, students used the Pythagorean theorem and the primary trigonometric ratios to find missing side lengths and angle measures in right triangles. They also investigated and worked with angles of elevation and depression (M4). In this unit, students will evaluate the primary trigonometric ratios for angles from 0° to 360° using the coordinate plane. They will also use the primary trigonometric ratios to solve problems involving oblique triangles, using the Law of Sines (including the ambiguous case) and the Law of Cosines. Achievement Indicators: T1.1 Sketch an angle in standard position, given the measure of the angle. In this unit, students will solve problems involving angles with measures between 0° and 360°. Negative rotational angles and co-terminal angles are not part of this outcome. They will be addressed in Mathematics 3200. T1.2 Determine the quadrant in which a given angle in standard position terminates. Introduce students to terminology such as the initial arm, terminal arm, vertex and standard position. They will sketch angles in standard position on the coordinate plane and identify the quadrant where the terminal arm lies. T1.3 Determine the reference angle for an angle in standard position. An understanding of reference angles will be critical when students compare the trigonometic ratio of an angle in standard position to the trigonometric ratio of its reference angle. It will also be useful when determining the exact trigonometric ratios for angles in standard position that are multiples of 30°, 45°and 60°, and solving equations of the form sin θ = a or cos θ = a. T1.4 Explain, using examples, how to determine the angles from 0° to 360° that have the same reference angle as a given angle. T1.5 Illustrate, using examples, that any angle from 90° to 360° is the reflection in the x-axis and/or the y-axis of its reference angle. To help students visualize the relationship between an angle sketched in standard position and its corresponding reference angle, provide students, individually or in small groups, with a copy of a circle sketched on grid paper. Give them a colored cut out of the right angle showing the corresponding angle sketched in standard position. 60° 60° Guide students using the following directions and questions: • 42 Place the right triangle over the angle on the circle. What do you notice about the two angles? They should verify the 60° angle is in standard position. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Authorized Resource Observation • Teachers create a model of the coordinate grid with an initial arm and terminal arm that can be moved physically. Ask students to demonstrate the placement of the terminal arm when given the measure of an angle between 0° and 360°. They should explore acute, obtuse, right, straight and reflex angles . Pre-Calculus 11 (T1.1) Paper and Pencil • Provide students with angles sketched in standard position from Quadrants II, III and IV, such as 150°, 210°, and 330°. Ask them to determine the related reference angle and explain why the reference angle for all three is the same. (T1.1, T1.2, T1.3, T1.4) • 2.1 Angles in Standard Position Student Book (SB): pp. 74-87 Teacher Resource (TR): pp. 57-64 Blackline Master (BLM): 2-3, 2-4 Ask students to determine the reference angle for each of the following angles: (i) 100° (ii) 250° (iii) 315° (T1.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 43 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T1 Continued ... Achievement Indicators: T1.3, T1.4, T1.5 Continued • Reflect the triangle across the y-axis into Quadrant II. • Determine the measure of the Quadrant II angle that is formed with the positive x-axis and the hypotenuse (terminal arm) of this triangle. 120° 60° • 60° Explain how the two angles are related. Students should recognize that 120° = 180° - 60° or θR = 180° - θ°. • Continue this activity in Quadrant III by asking students to rotate the 60° original angle 180° about the origin. Then ask students to reflect the original angle across the x-axis into Quadrant IV. Encourage students to explain their observations. Students should observe that in Quadrant III, the two angles have the relationship 240° = 180° + 60° or θR= 180° + θ°. In Quadrant IV, the angles are 300° = 360°- 60° or θR = 360° - θ°. Now that students have explored all quadrants, this would be a good opportunity to question them about the general properties of reference angles. Students should notice that the reference angle is always positive and measures between 0° and 90°. T1.6 Draw an angle in standard position given any point P(x, y) on the terminal arm of the angle. T1.7 Illustrate, using examples, that the points P(x, y), P(-x, y), P(-x, -y )and P(x, -y) are points on the terminal sides of angles in standard position that have the same reference angle. 44 Once students have sketched an angle in standard position and determined the quadrant in which it terminates, it is a natural extension to draw an angle given a point on its terminal arm. Students will determine which quadrant the terminal arm of the angle is located based on the point P(x, y) given. Students should be given an opportunity to graph a point and reflect it on the y-axis, on the x-axis and across the origin. The various reflections will illustrate how a point on the terminal arm of (1, 2), for example, reflects to become (-1, 2) in quadrant II, (-1, -2) in quadrant III and (1, -2) in quadrant IV. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Interview Authorized Resource • Pre-Calculus 11 Provide students with examples representing the endpoint of the terminal arm for angles sketched in standard position from Quadrants II, III and IV (e.g., (-3, 4), (-3, -4) and (3, -4) ). Ask them to explain why the related reference angle for all three is the same. (T1.1, T1.2, T1.3, T1.4) 2.1 Angles in Standard Position SB: pp. 74-87 TR: pp. 57-64 BLM: 2-3, 2-4 Observation • Ask students to explore the endpoint of the terminal arm of θ if it lies on the axis (ie., (1, 0)(0, 1), (-1, 0) and (0, -1)). (T1.6, T1.7) Performance • Ask students to create a “Four Door Book” foldable. On the front of the foldable, they should write the quadrants. On the inside of the foldable, students write the general “rule” for generating the reference angle in each quadrant and provide an example. Front Quadrant II Quadrant I Inside θR θR = 180° - θ Quadrant III Quadrant IV θR = 180° + θ Web Link The following site provides information on how to create different types of foldables. θR = 360° - θ http://mathnmind.com/PDF%20 Files/Pre-Algebra/dzf.pdf (T1.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 45 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T2 Solve problems, using the three primary trigonometric ratios for angles from 0° to 360° in standard position. [C, ME, PS, R, T, V] In Mathematics 1201, students applied the primary trigonometric ratios to angles between 0° and 90°. They will now explore angles between 0° and 360° using coordinates and reference angles. If θ is an angle in standard position, and point P(x, y) is a point on the terminal arm of angle θ, students will use the Pythagorean theorem to determine the length of the hypotenuse r. The three primary trigonometric ratios will be defined in terms of x, y and r. sin θ = opposite side hypotenuse = y r cosθ = adjacent side hypotenuse = x r tan θ = opposite side adjacent side = y x Students will work with exact trigonometric ratios which may require finding exact values for the hypotenuse. Special angles formed by the intersection of the x-axis and y-axis will also be investigated (i.e., quadrantal angles). Students will be expected to solve simple trigonometric equations of the form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1 , and tan θ = a, where a is a real number. Achievement Indicators: T2.1 Determine, using the Pythagorean theorem, the distance from the origin to a point P(x, y) on the terminal arm of an angle. An activity such as the following provides an opportunity for students to explore the trigonometric ratios given any point on the terminal arm of angle θ. Begin with the point (-4, -5). Continue the activity to ensure that endpoints from all quadrants are used. Use the following directions and questions to guide students: • T2.2 Determine the value of sin θ, cos θ, or tan θ given any point P(x, y) on the terminal arm of angle θ . Plot the given point P(-4, -5). What quadrant does this point lie in? • Construct the corresponding angle in standard position. • T2.3 Determine the sign of a given trigonmetric ratio for a given angle, without the use of technology, and explain. Drop a perpendicular to the x-axis creating a right triangle. Which value represents the adjacent side? Which value represents the opposite side? • How can you determine the exact length of the hypotenuse? • State the cosine, sine and tangent ratios associated with the angle. • What determines the sign of the ratio? Explain your reasoning. T2.4 Sketch a diagram to represent a problem. If θ is an angle in standard position, students should be able to summarize in which quadrant(s) the terminal arm of θ will lie if: (i) tan θ > 0 or tan θ < 0 (ii) sin θ > 0 or sin θ < 0 (iii) cos θ > 0 or cos θ < 0 46 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to complete the following table and suggest strategies for determining the sign (±) of the various ratios in each of the quadrants. They should share and compare their findings with the rest of the class. cos θ sin θ tan θ I II III IV (T2.4) 2.2 Trigonometric Ratios of Any Angle SB: pp. 88-99 TR: pp. 65-72 BLM: 2-3, 2-5 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 47 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T2 Continued... Achievement Indicators: T2.5 Determine, without the use of technology, the value of sin θ, cos θ, or tan θ given any point P(x, y) on the terminal arm of angle θ, where θ = 0 °, 90 °, 180 °, 270 °or 360 ° . Trigonometric ratios for angles whose measurements are 0°, 90°, 180°, 270° or 360° will now be explored. Ask students to choose an angle and then sketch it on the coordinate plane. Ask them the following questions using the 90° angle as a guide: T2.4 Continued • Where is the terminal arm? • Does this angle have a corresponding reference angle? • Can a reference triangle be drawn? Explain. Students should recognize that every angle drawn in standard position has a corresponding reference angle, except for quadrantal angles. While the unit circle is not formally introduced until Mathematics 3200, this would be an effective strategy to use when discussing quadrantal angles. The unit circle, where the radius equals one and the centre is (0, 0), can help students understand facts about the sine and cosine function. P(0, 1) P(x, y) 1 P(-1, 0) θ x y P(1, 0) P(0, -1) Encourage students to use the diagram to help them write their definitions for sine, cosine and tangent for each quadrantal angle. Since y y cosθ = xr = x1 = x and sin θ = r = 1 = y then P(x, y) = (cos θ, sin θ). Challenge students to further explore this activity using the points (3, 0), (0, 3), (-3, 0) and (0, -3). Ask them how the length of the terminal arm affects the values of the cosine and sine ratios. 48 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Observation Authorized Resource • Pre-Calculus 11 Ask students to plot the points (1,0), (0,1), (-1,0) and (0,-1), and evaluate, using technology, the values of the sine and cosine ratio for 0°, 90°, 180°, 270° or 360°. They should explain any patterns or relationships that appear relating the x and y coordinate of each endpoint to the value of each trigonometric ratio. (T2.4, T2.5) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 2.2 Trigonometric Ratios of Any Angle SB: pp. 88-99 TR: pp. 65-72 BLM: 2-3, 2-5 49 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T2 Continued ... Achievement Indicators: T2.6 Solve, for all values of θ, an equation of the form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1, and an equation of the form tan θ = a, where a is a real number. Previously, students were given an angle from 0° to 360° and asked to find the trigonometric value. They will now work backwards to find the missing angle when given an equation of the form sin θ = a or cos θ = a, where −1 ≤ a ≤ 1, and the equation of the form tan θ = a, where a is a real number. Use the following questions to guide them through the solution of an equation such as cos θ = 0.25: • Determine the quadrants which contain solutions. Are there any restrictions? • Determine the reference angle for the given value of a = 0.25. • Determine the measure of the related angles in standard position where 0° ≤ θ ≤ 360°. Students will be exposed to problems where they will have to rearrange the equation before they solve it. Ask them how they might solve an equation such as -2cos θ - 1 = 0 by comparing it to the linear equation -2x - 1 = 0. They should explain how the process is similar and how it is different. T2.7 Determine the exact value of the sine, cosine or tangent of a given angle with a reference angle of 30°, 45°or 60°. T2.4 Continued Students have calculated the trigonometric values for angles between 0° and 360°. They have also discovered relationships that exist between the angle and its reference angle. For example, sin θ = - sin(θR) where θ is an angle in the third quadrant. Reference angles of 30°, 45° or 60° occur frequently in problems. It is equally important for students to understand where the sine, cosine or tangent value of these angles come from. Geometric properties of right triangles containing 30°, 45° and 60° can be used to obtain the trigonometric values. To demonstrate the exact trigonometric value for θ = 45°, students can construct an isosceles triangle with the smallest side measuring 1 unit. They can then use the Pythagorean theorem to determine the length of the hypotenuse, 2 . 50 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • Ask students to solve the equations for 0° ≤ θ ≤ 360°. Pre-Calculus 11 (i) sin θ = 13 3 (ii) cosθ = − 7 2.2 Trigonometric Ratios of Any Angle (iii) tan θ = 3 SB: pp. 88-99 TR: pp. 65-72 BLM: 2-3, 2-5 (iv) cos θ = 0.8660 (v) sin θ = 0.7071 (T2.6) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 51 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T2 Continued ... Achievement Indicators: T2.7, T2.4 Continued Consider the following guiding questions: • What is the exact value of sin 45°? • What do you notice if you evaluate sin 45° with a calculator? • What do you notice about the values of sin 45° and cos 45°. Can you explain why they have the same value? • What is the value of tan 45°? Can you determine the value of tan 90° using the right triangle? Explain your reasoning. • What is the same and what is different about sin 45° and sin 225°? • What is the same and what is different about cos 45° and cos 135°? • What is the same and what is different about tan 45° and tan 135°? When determining the exact value of the trigonometric ratios of a given angle, students will sometimes result in a value where the denominator contains a radical. Consider the example, cos 45°= 12 . Students will not be expected to rationalize the denominator since the Radicals Unit is not done until later in this course. A similar strategy to demonstrate the exact trigonometric values for θ = 30° or θ = 60° is for students to construct an equilateral triangle with a side length of 2. Drawing an altitude from vertex A, students can then use the Pythagorean theorem to determine the length of the altitude, 3 . A 30° C 52 2 60° 1 B ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Divide the class into two groups. Individual students in one group will be given a card containing trigonometric ratios of special angles (i.e., sin 60°, cos 120°, etc). In the other group, students should be given associated exact values (ie., 23 , − 23 , etc). Ask students to find a partner to form a matching pair. (T2.7, T2.4) Journal • When evaluating sin 45°, ask students to explain why an exact value while 0.707 is approximate. Interview • 1 2 2.2 Trigonometric Ratios of Any Angle SB: pp. 88-99 TR: pp. 65-72 BLM: 2-3, 2-5 is called (T2.7, T2.4) Ask students to respond to the following questions when working with a 30°-60°-90° triangle. (i) Calculate the exact value of sin 30° and cos 30°. (ii) Calculate the exact value of sin 60° and cos 60°. (iii) Determine sin 30°, cos 30°, sin 60° and cos 60° using a calculator. What do you notice? (iv) Which is greater, sin 60° or sin 30°? Why? Note: The ratios are often left in irrational form in the resource. (i.e., sin 45° = 12 ) (v) What is the same and what is different about sin 30° and sin 150°? (vi) What is the same and what is different about cos 30° and cos 150°? (vii) What do you notice when you compare the exact value of tan 30° and tan 60° using the right triangle? (viii)What is the same and what is different about tan 30° and tan 330°? (T2.7, T2.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 53 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T2 Continued ... Achievement Indicators: T2.8 Describe patterns in and among the values of the sine, cosine and tangent ratios for angles from to 0° to 360°. T2.9 Solve a contextual problem, using trigonometric ratios. T2.4 Continued Encourage students to use mental math to determine the trigonometric function values whenever the terminal side makes a 30°, 45° and 60° angle with the x-axis. Students should be given an opportunity to explore the patterns in the sine, cosine and tangent ratios. They could use graphing technology and a table to record the points. Remind students to choose an appropriate increment for θ when graphing y = sin θ, y = cos θ and y = tan θ from 0° to 360°. Students will be expected to solve problems using the trigonometric ratios. Encourage students to draw a sketch of a diagram to help them gain a visual understanding of the problem. Consider an example such as the following: • 54 The arm of a crane used for lifting very heavy objects can move so that it has a minimum angle of inclination of 30° and a maximum of 60°. Use exact values to find an expression for the change in the vertical displacement of the end of the arm, in terms of the length of the arm, a. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Observation • Observe students graphing y = sin θ, y = cos θ and y = tan θ from 0° to 360°. Ask them to answer the following: (i) What is the maximum value of sin θ, cos θ and tan θ? the minimum value? (ii) Where does the maximum value occur? the minimum value? (iii) What are the x-intercepts and y-intercepts? Resources/Notes Authorized Resource Pre-Calculus 11 2.2 Trigonometric Ratios of Any Angle SB: pp. 88-99 TR: pp. 65-72 BLM: 2-3, 2-5 (iv) For what values of θ is the function positive? For what values is it negative? (v) What comparisons can you make between the sine and cosine function? (T2.8) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 55 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T3 Solve problems, using the cosine law and the sine law, including the ambiguous case. [C, CN, PS, R, T] In Mathematics 1201, students used the three primary trigonometric ratios to determine the side lengths and angle measures in right triangles (M4). Solving problems involving trigonometry is rarely limited to right triangles. In this unit, students will derive the Law of Sines and the Law of Cosines and utilize them in a number of problem-solving situations. Achievement Indicators: T3.1 Sketch a diagram to represent a problem that involves a triangle without a right angle. Students have been exposed to right-triangle trigonometry to solve problems involving right triangles. They will now solve oblique triangles. Encourage students to draw a diagram to help them gain a visual understanding of the problem. Consider the following example: A 10 T3.2 Solve, using primary trigonometric ratios, a triangle that is not a right triangle. 66° 44° B x C Ask students if this triangle can be divided into two right triangles and what strategies can be applied to find the indicated side length. They should recognize that this requires a multi-step solution. A strategy must be developed before a solution is attempted. Drawing an altitude from vertex A, students can use the primary trigonometric ratios and the Pythagorean theorem to solve for the unknown value. A 10 y cos44° = 10 ( y = 7.2) 10 2 = 7.2 2 + h 2 ( h = 6.9) h 66° 44° B y w x C tan 66° = 6.9 ( w = 3.1) w x = y+w Students will be introduced to methods that are more efficient when solving an oblique triangle, namely the sine law and the cosine law. 56 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to determine the length of AC in the following diagram: 2.3 The Sine Law SB: pp. 100-113 TR: pp. 73-80 BLM: 2-3, 2-7 (T3.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 57 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T3 Continued ... Achievement Indicator: T3.3 Explain the steps in a given proof of the sine law and cosine law. Students will prove the sine law and then use the law to solve triangles. The Law of Sines can be derived using the area formula of a right triangle. Consider the following diagram: B c h C A b Triangle ABC is not a right triangle. Therefore, students will draw an altitude from vertex B. The area formula for a right triangle is Area = 21 (base)(height). Ask students to write an expression for the height (h) using the sine ratio. Since sin A = hc , then h = c(sinA). Therefore, Area = 21 bc sin A . Ask students to repeat this procedure by drawing an altitude from the other vertices and writing an equation for the area of triangle ABC. B B a c c A a h h C A b C They should conclude that Area = 21 ac sin B and Area = 21 ab sin C Promote student discussion as to why the three area expressions are equivalent. Since 21 bc sin A = 21 ac sin B = 21 ab sin C , students can divide each expression by 21 abc to obtain the sine law. sin A sin B sin C = = a b c In other words, the sine law is a proportion that compares the ratio of each side of a triangle to its included angle. 58 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Provide students with a triangle and have them measure the side lengths and angles using a ruler and protractor. 2.3 The Sine Law A SB: pp. 100-113 TR: pp. 73-80 BLM: 2-3, 2-7 B C Measure Measure Calculate ∠A side a sin A a ∠B side b sin B b ∠C side c sinC c Ask students to answer the following questions: (i) What can you conclude regarding the ratios calculated above? (ii) Would your conclusion be valid if you were to use the reciprocal of the ratios? (T3.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 59 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T3 Continued ... Achievement Indicator: T3.4 Sketch a diagram and solve a problem, using the sine law. Students will apply the sine law to determine unknown lengths and angle measures in triangles. Encourage them to draw diagrams with both the given and unknown information marked. Ask students what information is needed to solve problems using the sine law. The Law of Sines involves a ratio of the sine of an angle to the length of its opposite side. Students should recognize that it will not work if no angle of the triangle is known or if one angle and its opposite side is not given. When working with the sine law, students sometimes incorrectly identify side and opposite angle pairs. To avoid this error, encourage them to use arrows on the diagram when identifying the angle and its opposite side. They could also encounter problems when they cross inA = sin 30° , for example, students multiply the ratios. When solving s 12 4 may incorrectly write 4sinA = sin 360°. Teachers should emphasize the use of brackets and write 4(sinA)= 12(sin 30°). Another common student error occurs when students try to solve a triangle given two angles and an included side and mistakenly think there is not enough information to use the sine law. Consider an example such as the following: C b A 43° 71° B c = 12 Students can use the property that the sum of the angles in a triangle is 180°. Therefore, the measure of ∠ C is 66°. They can then proceed to use the sine law to find the length of side AC. Encourage students to check the reasonableness of their answer. For example, since ∠ C is a little smaller than ∠ A, we expect the length of side AB to be a little shorter than the length of side CB. Students should also consider asking questions such as: Is the shortest side opposite the smallest angle? Is the longest side opposite the largest angle? 60 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Observation Authorized Resource • Pre-Calculus 11 Provide students with several practice problems using the sine law. As teachers observe students working through the problems, ask them the following questions: (i) What is the unknown? Is it an angle or a side? (ii) How can you isolate the unknown? 2.3 The Sine Law SB: pp. 100-113 TR: pp. 73-80 BLM: 2-3, 2-7 (iii) How can you complete the calculations? (iv) How do you know whether to determine the sine or the inverse of the sine? (v) Does your conclusion answer the question asked? (T3.4) Interview • A surveyor is located on one side of a river that is impossible to cross and only has a 100 m measuring tape and a sextant (used to measure angles) in his possession. Ask students to explain how the surveyor could use only these two tools and the Law of Sines to find the distance from point A to point C. (T3.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 61 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T3 Continued ... Achievement Indicator: T3.5 Describe and explain situations in which a problem may have no solution, one solution or two solutions. When using the sine law, students will be exposed to the following situations: (i) Provided with two angles and an included side (ASA) (ii) Provided with two angles and a non-included side (AAS) (iii) Provided with two sides and a non-included angle (SSA) – the ambiguous case The ambiguous case may cause difficulty for some students. Provided with two sides and an angle opposite one of those sides, students generally expect that one triangle will result. A numerical example, outlined below, could help students further investigate the Law of Sines and observe special situations where one, two or no triangle is in fact possible. Students are familar with the 30°-60°-90° triangle. C 2 a=1 30° A B Using C as the center of the circle, ask students to construct ∠ A = 30° and length AC = 10. Students should first notice that one triangle is possible where ∠ B =90° and a = 5. They can verify this using the Law of Sines. sin 30° 5 C 10 a=5 30° A B = sin10B sin B = 10(sin 30° ) 5 sin B = 1 ∠B = 90° Students should then consider the case where a < 5. Using geometry, draw an arc length a = 4. sin 30° 4 = sin10B sin B = 10(sin 30° ) 4 sin B = 1.25 Students should notice that no triangle is physically possible. If they do not attempt this approach, they may use the Law of Sines to try and solve this problem. This is an opportunity for students to review the maximum and minimum values of y = sin θ. Students should recognize that no triangle exists since the value of sin B > 1. 62 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to solve each of the following triangles using the Law of Sines (if possible) indicating which results in one triangle, two triangles or no triangle. (i) ∠ A = 45°, ∠ B = 60°, and a = 14 (ii) ∠ B = 40°, a = 12, b = 6 2.3 The Sine Law SB: pp. 100-113 TR: pp. 73-80 BLM: 2-3, 2-6, 2-7 (iii) ∠ A = 25°, a = 2, b = 3 (T3.5) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 63 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T3 Continued ... Achievement Indicators: T3.5 Continued T3.3 Continued Ask students to repeat this activity where a > 5. Using an arc length of a = 8, for example, the Law of Sines will yield two possible values for angle B (i.e., ∠B = 53.1° or ∠B = 126.9° ) and therefore, two possible triangles exist. Students will prove and use the cosine law to solve triangles. Ask students to consider the triangle ABC with side lengths a, b and c . Draw an altitude, h, from vertex C and let D be the intersection of AB and the altitude as shown in the figure below. If x is the length of AD, then students should recognize that BD = c - x. Guide students through the following process: C b A x a h D c-x B c • Use the Pythagorean theorem in ∆BCD : a2 = h2 + (c - x)2 • Expand the binomial: a2 = h2 + c2 - 2cx + x2 • Apply the Pythagorean theorem in ∆ADC (x2 + h2 = b2). What do the expanded binomial yield if b2 - x2 is substituted for h2. They should recognize the equation will result in a2 = b2 + c2- 2cx. • What primary trigonometric ratio can be used to determine the altitude? They should recognize that in ∆ACD, cos A = xb , resulting in x = bcosA. • Substitute x = bcosA into the equation a2 = b2 + c2- 2cx. What do you notice? (a2 = b2 + c2 - 2cbcosA) Students can then express the formula in different forms to find the lengths of the other sides of the triangles. 64 b2 = a2 +c2 -2accosB c2 = a2 +b2 -2abcosC ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Journal Authorized Resource • Pre-Calculus 11 Ask students to explain how the Law of Cosines validates the Pythagorean theorem if the included angle is 90°. (T3.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 2.4 The Cosine Law SB: pp. 114-125 TR: pp. 81-87 BLM: 2-3, 2-8 65 TRIGONOMETRY Trigonometry Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to T3 Continued ... Achievement Indicator: T3.6 Sketch a diagram and solve a problem, using the cosine law. The cosine law can be used to determine an unknown side or angle measure in a triangle. Continue to encourage students to draw diagrams with both the given and unknown information marked when solving problems. Ask them what information is needed to use the cosine law. They should consider why the cosine law is the only option to find the unknown angle if three sides are known or if two sides and the included angle are known. When three sides of a triangle are known, students will use the cosine law to find one of the angles. Some students may rearrange the equation to solve for a particular angle. Others may substitute the unknown values into the cosine law and then rearrange the equation to find the angle. It is important for them to recognize they have a choice when trying to find the second angle. They can either use the cosine law or the sine law. Students should notice the third angle can then be determined using the sum of the angles in a triangle. When solving triangles, encourage students to consider the following questions: • What is the given information? • What am I trying to solve for? • With the given information, should I use the sine law or the cosine law? Is there a choice? • Which form of the cosine law do I use to solve for an unknown side? Which form do I use to solve for an unknown angle? If students know two sides and a non-included angle, they can use the cosine law in conjunction with the sine law to find the other side. As an alternative, they could apply the sine law twice. Students have to be exposed to numerous examples to find the method that works best for them. When working with the cosine law, students sometimes incorrectly apply the order of operations. When asked to simplify a2 = 365 - 360cos70°, for example, they often write a2 = 5cos70°. To avoid this error, teachers should emphasize that multiplication is to be completed before subtraction. 66 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 TRIGONOMETRY General Outcome: Develop trigonometric reasoning. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 In the activity, Four Corners, students have to think about which method they would use to solve a triangle. Post four signs, one in each corner labelled sine law, cosine law, Pythagorean theorem, trigonometric ratios. Provide each student with one triangle. Instruct the students to make a decision as to which method they would use to find the missing angle or side and to stand in the corner where it is labelled. Once students are all placed, ask them to discuss why their triangle(s) would be best solved using that particular method. Sample triangles are given below: B a a A A 66° C C A 26 cm B 22 cm 31 cm x° 88° 32° 33 cm 21 cm B 72° x° C B A C www.k12pl.nl.ca/seniorhigh/ introduction/math2201/ classroomclips.html 35 cm C 92 cm Web Link In the Trigonometry clip, students participate in the activity Four Corners. They discuss the strategy to solve for the missing side lengths and angles of a given triangle. 22 cm 14 cm A B 2.4 The Cosine Law SB: pp. 114-125 TR: pp. 81-87 BLM: 2-3, 2-8 65 cm c 67 cm B A x° C 42 cm (T3.4, T3.6) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 67 TRIGONOMETRY 68 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Quadratic Functions Suggested Time: 12 Hours QUADRATIC FUNCTIONS Unit Overview Focus and Context In this unit, students will be introduced to standard form and vertex form of a quadratic function. They will investigate the characteristics of a quadratic function using features such as x-and y-intercepts, vertex, axis of symmetry, domain and range. Students will solve problems involving situations that can be modelled by quadratic functions. Outcomes Framework GCO Develop algebraic and graphical reasoning through the study of relations. SCO RF3 Analyze quadratics of the form y = a(x - p)2 + q and determine the: • vertex • domain and range • direction of opening • axis of symmetry • x-and y-intercepts. SCO RF4 Analyze quadratic functions of the form y = ax2 + bx + c to identify characteristics of the corresponding graph, including: • vertex • domain and range • direction of opening • axis of symmetry • x-and y-intercepts and to solve problems. 70 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Mathematics 2200 Mathematics 3200 Algebra and Number Relations and Functions Relations and Functions AN4 Demonstrate an understanding of the multiplication of polynomial expressions (limited to monomials, binomials and trinomials) concretely, pictorially and symbolically. RF3 Analyze quadratics of the form y = a(x - p)2 + q and determine the: [CN, R, V] AN5 Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially and symbolically. [C, CN, R, V] • vertex • domain and range • direction of opening • axis of symmetry • x- and y-intercepts. [CN, R,T,V] RF4 Analyze quadratic functions of the form y = ax2 + bx + c to identify characteristics of the corresponding graph, including: • vertex • domain and range • direction of opening • axis of symmetry • x-and y-intercepts RF11 Demonstrate an understanding of factoring polynomials of degree greater than 2 (limited to polynomials of degree ≤ 5 with integral coefficients). [C, CN, ME] RF12 Graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5). [C, CN, T, V] and to solve problems. [CN, PS, R, T, V] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 71 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF3 Analyze quadratics of the form y=a(x - p)2 + q and determine the: • vertex • domain and range • direction of opening • axis of symmetry • x-and y-intercepts. [CN, R,T, V] Achievement Indicator: RF3.1 Explain why a function given in the form y = a(x - p)2 + q is a quadratic function. In Mathematics 1201, students were introduced to functional notation through work with linear functions. They were introduced to the terms “relation” and “function” and determined if a relation was a function (RF2). They also determined the domain value of a linear function given a range value and vice versa (RF8). The domain and range of a graph was written using interval notation or set notation (RF1). In this unit, students will be introduced to quadratic functions. They will first examine quadratic functions expressed in vertex form, y = a(x - p)2 + q, and determine the x-and y-intercepts, vertex, axis of symmetry, direction of opening, and domain and range. They will also sketch graphs using the characteristics above as well as transformations. Using inductive reasoning, students will study the effect of changing the parameters a, p and q on the shape of the graph and on the number of x-intercepts. Before students are exposed to the vertex form of a quadratic, they need to become familar with the shape of a quadratic function and how to identify a quadratic function. The terms “quadratic” and “parabola” are new to students. This will be their first exposure to functions that are non-linear. Students should have an opportunity to investigate what makes a quadratic function. Ask studets to turn to a partner to multiply two linear equations or square a binomial of the form ax + b. Consider the examples y = (x + 1)(x - 4) and y = (3x - 2)2. Ask students what they notice in terms of the degree of the polynomial and if this is similar to expanding functions of the form y = a(x - p)2 + q. Projectile motion can be used to explain the path of a baseball or a skier in flight. To help students visualize the motion of a projectile, toss a ball to a student. Ask students to describe the path of the ball to a partner and sketch the path of the height of the ball over time (the independent axis represents time and the dependent axis represents height of the ball). Encourage them to share their graphs with other students. Ask them to think of other examples that might fit the diagrams of parabolas that open upward or downward. They should explain why they think quadratic relations represent functions. Characteristics of the resulting parabola, such as the vertex and axis of symmetry, should be discussed. It is important for them to recognize that each point on a parabola has a corresponding point on its mirror image, resulting in a parabolic shape. Students may conclude that the graph resembles a U-shape. It is important to note, however, that obtaining a U-shaped graph does not guarantee the function is quadratic. Other functions may produce a similarly shaped graph. The graph of y = x4, for example, may be misinterpreted as being parabolic if not examined closely. 72 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • Provide the following table with a set of statements. Ask students to describe the reasoning they used to decide whether each statement is true or false. Polynomial Classification True or False Explain/Justify Function y = 5(x + 3) Linear 2 y = 5(x + 3) Quadratic 2 y = 5 (x + 3) Quadratic y = 5x(x + 3) Linear y = (5x + 1)(x + 3) Quadratic y = 5(x + 3)2 + 2 Quadratic (RF3.1) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Pre-Calculus 11 3.1 Investigating Quadratic Functions in Vertex Form Student Book (SB): pp.142-162 Teacher Resource (TR): pp.103112 BlackLine Master (BLM): 3-3, 3-4 73 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF3 Continued ... Achievement Indicators: RF3.2 Compare the graphs of a set of functions of the form y = ax2 to the graph of y = x2, and generalize, using inductive reasoning, a rule about the effect of a. As students begin varying the parameters of a quadratic function, they will first explore quadratic functions in vertex form. The characteristics of the quadratic function, y = a(x - p)2 + q, should be developed through an investigation where the parameters a, p and q are manipulated individually. Technology, such as graphing calculators, FX Draw, Graphmatica, Geometer’s Sketchpad, Winplot, Geogebra, or other suitable graphing software, should be used, as students’ graphing abilities for quadratics would be limited to the use of a table of values at this point. Students will first investigate the effect of changing the value of a by comparing quadratic functions y = x2 and y = ax2. As they compare the graphs of y = 2x2 and y = -2x2 to the graph of y = x2, use prompts such as the following to promote student discussion: • What happens to the direction of opening of the quadratic if a < 0 and a > 0? • Is the shape of the parabola affected by the parameter a? • Are some graphs wider or narrower when compared to the graph of y = x2 ? • What is the impact on the graph if a = 0? • What effect does varying a have on the vertex? Encourage students to pay particular attention to the points and how they change. For example, the point (2,4) on the graph of y = x2 changes to the point (2,8) on the graph of y = 2x2. 2 Similarly, students can compare the graphs of y = 21 x and y = − 21 x 2 to the graph of y = x2. RF3.3 Compare the graphs of a set of functions of the form y = (x - p)2 to the graph of y = x2, and generalize, using inductive reasoning, a rule about the effect of p. RF3.4 Compare the graphs of a set of functions of the form y = x2+ q to the graph of y = x2, and generalize, using inductive reasoning, a rule about the effect of q. 74 Use graphing technologies to examine the effects of manipulating the values of p. Students will investigate the effect of changing the value of p by comparing quadratic functions y = x2 and y = (x - p)2. They could describe how the graphs of y = (x - 3)2 and y = (x + 3)2 compare to the graph of y = x2. Ask questions such as: • How does each graph change when compared to y = x2? • What effect does varying p have on the vertex? Similarly, students will examine the effects of manipulating the value of q. They will investigate the effect of changing the value of q by comparing quadratic functions y = x2 and y = x2 + q. They could describe how the graph of y = x2 + 2 and y = x2 - 2 compares to the graph of y = x2. Ask students what effect does varying q have on the vertex. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Ask students to draw a tile from a bag containing various values for the parameter a. They will then proceed to the board or on the wall of a classroom to complete a table, such as the one shown below, indicating the effect of a on the graph of y = x2. Ask students to discuss the results. Be sure to include positive and negative values for a, as well as integers and rational numbers. Value of a Function a=2 y = 2x2 a = − 12 y = − 21 x 2 Opening Up or Down up down 3.1 Investigating Quadratic Functions in Vertex Form SB: pp.142-162 TR: pp.103-112 BLM: 3-3, 3-4 Narrower or Wider narrower wider (RF3.2) Paper and Pencil • Ask students to complete the webbing to describe the effects of the parameters a, p and q on the quadratic function y = a(x - p)2 + q. y = a ( x − p) + q 2 a p q (RF3.2, RF3.3, RF3.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 75 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF3 Continued ... Achievement Indicators: RF3.5 Determine the coordinates of the vertex for a quadratic function of the form y = a(x - p)2 + q, and verify with or without technology. RF3.6 Generalize, using inductive reasoning, a rule for determining the coordinates of the vertex for quadratic functions of the form y = a(x - p)2 + q. In Grade 9, students graphed linear relations using a table of values (9PR2). They also investigated line symmetry (9SS5). In Mathematics 1201, students used technology and/or a table of values to graph linear functions (RF6). They will now extend these concepts to determine the vertex and axis of symmetry of the graph of a quadratic function. When graphing the quadratic function y = 3(x + 2)2 - 1, for example, students can create the following table of values by hand or through the use of graphing technology. x y -4 11 -3 2 -2 -1 -1 2 0 11 Promote student discussion by asking questions such as the following: • What is the connection between the axis of symmetry of the graph and the vertex? • What is the equation of the axis of symmetry? • What is the connection between the maximum or minimum point on the graph and the vertex? • How can the vertex be obtained directly from the vertex form of the quadratic function? The goal is for students to recognize that in a table of values, the vertex will be the point with the unique y-value. It is the point at which the graph of the quadratic changes direction, and it is the maximum or minimum point on the graph. Whether students use a graph or a table of values, they should be able to identify the symmetry of the graph since the points on the parabola that share the same y-coordinate are equidistant from the vertex. Students should also realize that the axis of symmetry of a parabola is the vertical line of symmetry that passes through the vertex and has equation x = p. 76 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Observation Authorized Resource • Ask students to graph y = 3(x - 2)2 + 1 using a table of values or graphing technology. They should compare the given graph to each of the following graphs and discuss the similarities and differences between the two. Pre-Calculus 11 (i) y = -3(x - 2)2 + 1 SB: pp.142-162 TR: pp.103-112 BLM: 3-3, 3-4 (ii) y = 3(x - 3)2+ 1 2 1 (iii) y = − 3 ( x − 2) + 4 3.1 Investigating Quadratic Functions in Vertex Form 2 1 (iv) y = 3 ( x − 2) + 1 (v) y = 21 ( x + 3)2 + 2 (vi) y = -2x2 + 1 (vii) y = 3x2 + 1 (RF3.1, RF3.2, RF3.3, RF3.4, RF3.5) Journal • Your friend has missed the class on determining the vertex for any quadratic function of the form y = a(x - p)2 + q and asks you to explain how this is done. Ask students to write a paragraph, with examples, explaining how this is done. (RF3.6) Web Link The following site provides a simulation that models changes in a, p, and q. (Note: the variables h and k are used in place of p and q). http://www.mathopenref.com (keyword: Quadratic Function Explorer) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 77 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF3 Continued ... Achievement Indicator: RF3.7 Sketch the graph of y = a(x - p)2 + q, using transformations, and identify the vertex, domain and range, direction of opening, axis of symmetry and x- and y-intercepts. Students previously investigated the effects on the graph of changing the values of a, p, and q in the vertex form of a quadratic function. They also found the vertex directly from the quadratic function. In Mathematics 1201, students graphed linear relations by plotting the x-and y-intercepts and found the domain and range of various relations (RF8 and RF1). They should recognize that all non-contextual quadratic functions have a domain of {x| x Î Â}, whereas the range depends on the vertex and the direction of opening. Ask students questions about their observations as they graph and analyze quadratic functions. They should consider the following: • Why is the domain the set of all real numbers when only some points are plotted from the table of values? • How is the range related to the direction of opening? Students should realize that a negative value of a will indicate that the range is less than or equal to q while a positive value of a will indicate that the range is greater than or equal to q. Students can consolidate their learning by sketching the graph of y = a(x - p)2 + q using transformations for various values of a, p, and q. They can apply the change in width using the value of a by selecting the vertex and two other points on the graph of y = x2. They can then use the values of p and q to translate the graph. Consider the function y = 3(x - 5)2 + 1. The a value, 3, results in a narrower graph. The points (0, 0), (-1, 1) and (1, 1) on the graph y = x2 will transform to (0, 0), (-1, 3), (1, 3) on the graph of y = 3x2. Students will then translate the graph using the horizontal translation (5 units) and the vertical translation (1 unit). The points transform to (5, 1), (4, 4), and (6, 4). Students should note the vertex and symmetry of the graph and other features such as the domain, range and intercepts. 2 y y=x 2 2 y = 3x y = 3(x – 5) + 1 8 6 4 2 -2 2 4 6 8 x -2 78 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Observation • Students can work in pairs to construct the graphs of a number of quadratic functions of the form y = a(x - p)2 + q, using transformations for various values of a, p and q. Each group is provided with a handout containing a grid and the function. Place the handout inside a sheet protector so each group can graph a different function using an erasable marker. Ask questions to the group related to the characteristics of the quadratic function (vertex, direction of opening, x-and y-intercepts, domain and range). When completed, ask students to erase the function and pass it along to another group. Repeat the activity, providing students the opportunity to construct a number of graphs. y 10 8 6 4 2 • 2 4 6 Pre-Calculus 11 3.1 Investigating Quadratic Functions in Vertex Form SB: pp.142-162 TR: pp.103-112 BLM: 3-3, 3-4 Web Link www.k12pl.nl.ca/seniorhigh/ introduction/math2200/ classroomclips.html 2 y = 2( x – 1) + 3 -10 -8 -6 -4 -2-2 -4 -6 -8 -10 Authorized Resource x 8 10 (RF3.7) The Frayer Model clip related to Characteristics of Quadratic Functions demonstrates students finding the different representations of a quadratic function and completing a puzzle. Students can work in pairs to complete the following quadratic puzzle investigating the characteristics and graphs of various quadratic functions of the form y = a(x - p)2 + q. They should work with 20 puzzle pieces (4 complete puzzles consisting of a function and four related characteristics) to correctly match the characteristics with each function. A sample is shown below. Vertex (2,4) { y | y ≤ 4} 2 y = -3( x – 2) + 4 y 10 8 Axis of Symmetry x=2 6 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 x -4 -6 -8 -10 (RF3.5, RF3.6, RF3.7) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 79 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF3 Continued... Achievement Indicators: RF3.8 Explain, using examples, how the values of a and q may be used to determine whether a quadratic function has zero, one or two x-intercepts. RF3.9 Write a quadratic function in the form y = a(x - p)2 + q for a given graph or a set of characteristics of a graph. Teachers could use a variety of examples to demonstrate the six different combinations of a and q shown in the table below. The use of graphing technologies or simulation software is recommended here. Value of a positive positive positive negative negative negative Value of q positive negative 0 0 positive negative Number of x-intercepts 0 2 1 1 2 0 Given a graph of a quadratic function, students will determine the quadratic function in the form y = a(x - p)2 + q. They can determine the vertex directly from the graph and hence, the values of p and q. When determining the value of a, students can substitute a point (x, y) that is on the parabola into the quadratic function and solve for a. Sometimes a description of the characteristics of the graph are given rather than the graph. The vertex is given, as a maximum or minimum value, and at least one other point. Using this information, students will determine the quadratic function in vertex form. 80 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Performance • Ask students to participate in the following card game. Place three headings labelled “No x-intercepts”, “One x-intercept”, and “Two x-intercepts” on the classroom wall. Make a set of cards for the values of a and q. For example, a = 3, -8, 2.67, 3 4 q = 0, -10, 11, −24 7 Pre-Calculus 11 3.1 Investigating Quadratic Functions in Vertex Form SB: pp.142-162 TR: pp.103-112 BLM: 3-3, 3-4 One student will select a card from the “a” deck and a second student will select a card from the “q” deck. The two students then collaborate to determine under which heading the two numbers should be placed. At the end of the activity, they should determine if any of the pairs of cards are misplaced. (RF3.8) Web Link Paper and Pencil • Ask students to determine the vertex form of the quadratic function from the given table of values. The Describe and Draw clip related to Characteristics of Quadratic Functions demonstrates (RF3.9) students graphing a quadratic function by listening to the Ask students to determine the vertex form of the quadratic information presented by their function given the following information. partner. (i) Range is {y / y ≤ 3, y∈ \ } and the x-intercepts are -2 and 4. x y • www.k12pl.nl.ca/seniorhigh/ introduction/math2201/ classroomclips.html -6 1 -5 3 -4 1 -3 -5 -2 -15 -1 -29 0 -47 (ii) Equation of the axis of symmetry is x = 2, the minimum value of y is -5, and the y-intercept is 3. (RF3.9) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 81 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF4 Analyze quadratic functions of the form y = ax2 + bx + c to identify characteristics of the corresponding graph, including: • vertex • domain and range • direction of opening • axis of symmetry • x-and y-intercepts and to solve problems. In this unit, students will examine quadratic functions expressed in standard form y = ax2 + bx + c and determine the x- and y-intercepts, vertex (as a maximum or minimum point), axis of symmetry, direction of opening, and domain and range. They will also sketch the graph of a quadratic function using the characteristics above. Students will study the effect of changing the parameters a, b, and c of the equation on the shape of the graph. They will also solve problems involving quadratic functions. [CN, PS, R, T, V] Achievement Indicators: RF4.1 Determine the characteristics of a quadratic function given in the form y = ax2 + bx + c, and explain the strategy used. RF4.2 Sketch the graph of a quadratic function given in the form y = ax2 + bx + c . The characteristics of the quadratic function, y = ax2 + bx + c, should be developed through an investigation where the parameters a, b, and c are manipulated individually. This should be completed through the use of technology. Ask students what parameters affect the shape of the graph, the direction of opening, and the y-intercept. Using the graph as a visual, ask students to discuss the following: • • • • maximum or minimum value domain and range equation of the axis of symmetry x-intercepts Students should first be exposed to quadratics for which the vertex has integer coordinates to make it easier to identify characteristics precisely. Students may use the formula x = − 2ba to find the equation of the axis of symmetry for quadratic functions of the form y = ax2 + bx + c. This value can then be used to determine the x-coordinate of the vertex. The formula should be developed through a student-based activity similar to the one outlined below. Provide students with a table containing quadratic functions written in standard form and the vertex of each corresponding parabola. Students should complete the table for the values of a, b and c along with the value of x = − 2ba . Function Vertex y = x2 - 4x + 7 (2, 3) Equation of Axis of Symmetry a b c − 2ba y = -2x2 - 16x - 34 (-4, -2) y = 3x2 - 6x + 10 82 (1, 7) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Performance • Ask students to work in groups for this activity. Each group should be given a pair of dice (or they can create their own). On the first die, there will be 2 a values, 2 b values, and 2 c values. On the second die, there will be various rational numbers. Students should roll the dice and state the effect on y = x2 of changing the given parameter to the given number. (RF4.1, RF4.2) Pre-Calculus 11 3.2 Investigating Quadratic Functions in Standard Form SB: pp.163-179 TR: pp.113-124 BLM: 3-3, 3-5 Interview • Ask students to explain how they could determine whether a quadratic function has either a maximum or minimum value without graphing. Note: In Mathematics 1204, standard form would have been referred to as general form. (RF4.1) Paper and Pencil Web Links • The following site provides an animation of how the shape of the graph is affected by the parameters a, b, and c. Ask students to complete the following webbing to describe the effects of the parameters a, b and c on the quadratic function y = ax2 + bx + c. y = ax 2 + bx + c a b http://www.livephysics.com/ simulations/math/quadraticequation-graph.html c The following is an interactive tutorial where students can vary the values of a, b, and c and its impact on the shape of the graph. http://www.analyzemath.com/ quadraticg/quadraticg.htm (RF4.1, RF4.2) • Ask students to write an email message explaining how changing the parameter a will affect the graph of the function y = ax2 + bx + c . (RF4.1, RF4.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 83 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF4 Continued ... Achievement Indicators: RF4.2 Continued RF4.3 Explain the reasoning for the process of completing the square as shown in a given example. Students should discover that the value of − 2ba is the x-coordinate of the vertex and its connection should be made to the equation of the axis of symmetry. The y-coordinate of the vertex can be found by substituting the x-coordinate into the quadratic function. Note that the y-coordinate of the vertex can also be found using the formula 2 y = 4 ac4−ab . This formula could be developed through an activity similar to the one described previously. In Mathematics 1201, students represented a quadratic expression using algebra tiles. They also demonstrated, through the use of algebra tiles, an understanding of the multiplication of polynomial expressions (limited to monomials, binomials and trinomials), common factoring and trinomial factoring (AN4, AN5). Students should be given an opportunity to understand the process of creating perfect square trinomials and the patterns formed. The method of completing the square is one method students will use when they rewrite a quadratic equation from standard form to vertex form. The perfect square binomial (x - p)2 is part of the quadratic function in vertex form. Algebra tiles can be used to visualize how a perfect square trinomial can be formed. Students should first be exposed to examples where a = 1 and c = 0. Consider the following example: Ask students to model x2 + 8x. The goal is to find a number c to create a perfect square trinomial x2 + 8x + c. Students will use the algebra tiles to create a square. Ask students why the number of tiles must be split evenly. Continue to use leading questions, such as the following, to promote discussion: • What tiles must be added to complete the square? 16 unit tiles 84 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Observation • Ask students to graph the following using a table of values or graphing technology. (i) (ii) (iii) (iv) (v) y = x2 y = x2 + 2 y = x2 + 5 y = x2 - 3 y = x2 - 4 3.2 Investigating Quadratic Functions in Standard Form SB: pp.163-179 TR: pp.113-124 BLM: 3-3, 3-5 As teachers observe students’ work, ask them to explain the changes in the parabola as the parameter c is manipulated. (RF4.1, RF4.2) Interview • Pre-Calculus 11 3.3 Completing the Square SB: pp.180-200 TR: pp.125-133 BLM: 3-3, 3-6 Using algebra tiles, ask students to model and explain the process of completing the square. The teacher photographs the group as they participate in the activity. After a few classes, give students their photograph and ask them to describe what they were doing in the picture. They should write about the activity under the photograph, describing what they were doing and what they learned as a result. (RF4.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 85 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF4 Continued ... Achievement Indicator: RF4.3 Continued • What is the expression that represents the new completed square? • What is the relationship between the coefficient of the linear term and the constant term? • What is the trinomial written as the square of a binomial? Students should realize 16 unit tiles have been added. Since the side length of the square is represented by x + 4, the area is (x + 4)(x + 4). This perfect square trinomial x2 + 8x + 16, can be rewritten as (x + 4)2. Continue to work with various examples, such as x2 + 2x, and x2 + 6x, to give students an opportunity to describe the pattern that exists. The algebra tile method illustrates the constant term is half the coefficent of the linear term squared. When modelling with algebra tiles, it is important for teachers to choose a value of b that is even and relatively small. Before students move from the concrete representation to the symbolic method give students an opportunity to work with expressions where a > 0 and c = 0. Consider the example: Ask students to model 2x2 + 8x using algebra tiles and why the tiles should be arrange into two equal parts. x2 + 4x x2 + 4x Ask students what tiles must be added to complete the square. 4 unit tiles 4 unit tiles Students will write the expression that represents the new completed square. They should recognize that 4 unit tiles were added to each diagram creating two perfect square trinomials, each of which can be written as x2 + 4x + 4 or (x + 2)2. Since there are two squares in the model, the total area is 2(x2 + 4x + 4) or 2(x + 2)2. This visual allows students to gain a better understanding as to why the value of b must be divided by a before the process of completing the square takes place. That is, 2x2 + 8x = 2(x2 + 4x). 86 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • • Ask students to complete the square for y = ax2 + bx + c to show 2 that the coordinates of the vertex are − 2ba , 4 ac4−a b . ) ( (RF4.1, RF4.3) Pre-Calculus 11 3.3 Completing the Square SB: pp.180-200 TR: pp.125-133 BLM: 3-3, 3-6 Ask students to complete a table investigating the relationship between the value of b and c when completing the square. They should share their findings with the class. A sample is shown below. Binomial of the form x2 + bx x2 + 6x What was added to complete the square? 9 Modeled with algera tiles Perfect square x2 + 6x + 9 Factored and expanded form Value of coeffcicent c How are b and c related? (x + 3)(x + 3)= x2 + 6x + 9 9 c = ( 21 b )2 (RF4.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 87 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF4 Continued ... Achievement Indicator: RF4.4 Write a quadratic function given in the form y = ax2 + bx + c as a quadratic function in the form y = a(x - p)2 + q by completing the square. This would be an opportunity for students to extend their work with algebra tiles to convert an equation from standard form to vertex form using the process of completing the square. They will then convert between the two forms algebraically. Use algebra tiles to model the function y = x2 + 6x + 7. Students will continue to split the tiles evenly and add tiles to form a square leaving the constant term alone. x2 + 6x 7 unit tiles It is important for students to notice that by adding nine extra tiles, the quadratic function has changed. Therefore, students will need to add 9 opposite tiles to balance the function. x2 + 6x+ 9 7-9 The function is y = (x2 + 6x + 9) + (7 - 9). They should recognize when a number is added to form a perfect square, its opposite value is also added to keep the original expression unchanged. Students can then rewrite this as y = (x + 3)2 - 2. This visual representation allows students to observe patterns and then record their work symbolically. Ask students why it is to their advantage to write an equation in vertex form. Students should emphasize that the coordinates of the vertex (p, q) can be directly determined when written in this form. Hence, the maximum or minimum point is known. Students should first be exposed to quadratic functions where the leading coefficient is 1 and then progress to examples where a ≠ 1. 88 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Ask students to work in pairs for this activity. Give each pair of students a quadratic equation in standard form and ask them to rewrite the equation in vertex form. Ask one student to write the first line of the solution and then pass it to the second student. The second student will verify the workings to determine if an error is present. If there is an error present, the student will correct it and then write the second line of the solution and pass it along to their partner. This process continues until the solution is complete. 3.3 Completing the Square SB: pp.180-200 TR: pp.125-133 BLM: 3-3, 3-6 (RF4.4, RF4.5) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 89 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF4 Continued ... Achievement Indicators: RF4.5 Identify, explain and correct errors in an example of completing the square. Common errors occur when converting a quadratic function from standard form to vertex form. A quadratic function where a ≠ 1, for example, sometimes causes difficulty for students. Consider the following example and discuss with students the possible errors that may occur: y = -3x2 + 18x - 23 y = -3(x2 - 6x) - 23 y = -3(x2 - 6x + 9) - 23 + 27 y = -3(x - 3)2 + 4 • The common factor (-3) is not factored out from both the quadratic and linear terms. • There is an incorrect sign on the linear term when a negative leading coefficient is factored out. • The constant term inside the parentheses is doubled instead of squared. • When a perfect square is created, the constant term inside the parentheses is not multiplied by the common factor to produce the compensated term. • RF4.6 Verify, with or without technology, that a quadratic function in the form y = ax2 + bx + c represents the same function as a given quadratic function in the form y = a(x - p)2 + q. The perfect square trinomial is incorrectly factored. When converting a quadratic function from vertex form to standard form, students will expand the perfect square trinomial and then combine like terms. Students should also verify that a quadratic function in standard form represents the same function in vertex form. They can use the method of completing the square to compare the functions or compare the graphs of both functions. Ask students what features of the graph must be the same. They must ensure that both functions have the same vertex and one other point (e.g., the y-intercept) on both graphs. As an alternative, an algebraic combined with a graphical approach could be used. Ask students to graph one of the functions. They should choose three points from the graph and substitute them into the other form of the function to verify the points satisfy the function. Promote student discussion as to why three points must be used and not two points. 90 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • In groups of two, ask students to play the Domino Game. Provide students with 10 domino cards. One side of the card contains a quadratic function in standard form, while the other side contains a quadratic function in vertex form. Ask each group to lay the dominos out such that the standard form on one card will match with the correct vertex form on another. They will eventually form a complete loop, with the first card matching the last card. Some sample cards are shown below: 2 y = ( x + 5) – 3 2 y = 2( x – 3) + 5 2 y = 2x – 12 x + 23 Pre-Calculus 11 3.3 Completing the Square SB: pp.180-200 TR: pp.125-133 BLM: 3-3, 3-6 Web Link The following site provides examples where students write the quadratic equation for parabolic shapes. 2 y = – x – 2x + 2 http://www.thefutureschannel.com/ algebra/windsail_design.php 2 y = – ( x + 1) + 3 2 y = x + 10 x + 22 (RF4.6) • In groups of two, ask students to move around the classroom to various stations where solutions have been posted outlining the process of completing the square. At each station the solution could contain one or more errors that the group has to identify and then produce correct solutions for each of the problems. (RF4.5) APPLIED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 91 QUADRATIC FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF4 Continued ... Achievement Indicators: RF4.7 Write a quadratic function that models a given situation, and explain any assumptions made. RF4.8 Solve a problem, with or without technology, by analyzing a quadratic function. 92 Students will be expected to write a quadratic function that models a situation and then solve the problem. They will be exposed to problems where they will determine the maximum or minimum value. Contextual problems could involve maximum revenue, finding the maximum possible area, the maximum height, or a minimum product. It is important for students to understand the terminology that is being used in the problems. They should make the connection between the maximum or minimum value, and the y-value of the vertex. MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to answer the following : (i) A stream of water from a fountain forms a parabolic shape. Given the spout on the fountain is 5 cm high and the maximum height reached by the water is 14 cm at a distance of 6 cm from the spout, what is the height of the water when it is 8 cm from the spout? 3.3 Completing the Square SB: pp.180-200 TR: pp.125-133 BLM: 3-3, 3-6 (ii) A student makes and sells necklaces at the beach during the summer months. The material for each necklace costs her $6.00 and she has been selling about 20 per day at $10.00 each. She has been wondering whether or not to raise the price, so she takes a survey and finds that for every dollar increase she would lose two sales a day. What price should she set for the necklaces to maximize profit? (iii) Find the dimensions of the rectangle of maximum area that can be inscribed in an isosceles triangle of altitude 8 and base 6. (Hint: Use similar triangles to express the height of the rectangle in terms of its base). 8 6 (RF4.7, RF4.8) APPLIED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 93 QUADRATIC FUNCTIONS 94 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Quadratic Equations Suggested Time: 14 Hours QUADRATIC EQUATIONS Unit Overview Focus and Context In this unit, students will extend their factoring skills and use a variety of strategies to determine the roots of quadratic equations. They will make the connection that the x-intercepts of the graph or the zeros of the quadratic function correspond to the solutions, or roots, of the quadratic equation. Students will explore various algebraic methods for solving quadratic equations including factoring, completing the square and the quadratic formula. They will use the discriminant to determine the number of real roots fro quadratic equations. Contextual problems will also be solved by modelling a situation with a quadratic equation. Outcomes Framework GCO Develop algebraic and graphical reasoning through the study of relations. SCO RF1 Factor polynomial expressions of the form: • ax2 + bx + c, a ≠ 0 • a2x2 - b2y2, a ≠ 0, b ≠ 0 • a(f(x))2 + b(f(x)) + c, a ≠ 0 • a2(f(x))2 - b2(g(y))2, a ≠ 0, b ≠ 0 where a, b and c are rational numbers. SCO RF5 Solve problems that involve quadratic equations. 96 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC EQUATIONS Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Mathematics 2200 Mathematics 3200 Algebra and Number Relations and Functions Relations and Functions AN5 Demonstrate an understanding of common factors and trinomial factoring, concretely, pictorially and symbolically. [C, CN, R, V] RF1 Factor polynomial expressions RF10 Solve problems that involve of the form: exponential and logarithmic equations. 2 • ax + bx + c, a ≠ 0 [C, CN, PS, R] • a2x2 - b2y2, a ≠ 0, b ≠ 0 • a(f (x))2 + b(f (x)) + c, a ≠ 0 • a2(f (x))2 - b2(g (y))2, a ≠ 0, b ≠ 0 RF11 Demonstrate an where a, b and c are rational understanding of factoring numbers. polynomials of degree greater than 2 (limited to polynomials of degree [CN, ME, R] ≤ 5 with integral coefficients). RF5 Solve problems that involve quadratic equations. [C, CN, ME] [C, CN, PS, R, T, V] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 97 QUADRATIC EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF5 Solve problems that involve quadratic equations. [C, CN, PS, R, T, V] In Mathematics 1201, students factored differences of squares, perfect square trinomials and polynomials of the form x2 + bx + c and ax2 + bx + c (AN5). In the previous unit, students were introduced to quadratic functions expressed in standard form and vertex form. They sketched graphs using characteristics such as x-and y-intercepts, vertex, axis of symmetry and domain and range. Students also converted a quadratic function in standard form to vertex form using the process of completing the square. In this unit, students will solve quadratic equations using various strategies, including factoring, completing the square, applying the quadratic formula or using a graph. Achievement Indicators: RF5.1 Explain, using examples, the relationship among the roots of a quadratic equation, the zeros of the corresponding quadratic function and the x-intercepts of the graph of the quadratic function. Students have been exposed to the graph of a quadratic function and the points where a parabola crosses the x-axis. They are aware a quadratic function can have zero, one or two x-intercepts. When solving a quadratic equation of the form ax2 + bx + c = 0, students can graph the corresponding quadratic function and determine the x-intercepts. They can use a table of values or graphing technology to make the connection between the x-intercepts of the graph and the roots of the quadratic equation. RF5.2 Solve a quadratic equation of the form ax2 + bx + c = 0 by using strategies such as: • determining square roots • factoring • completing the square • applying the quadratic formula • graphing its corresponding function. It is important for students to distinguish between the terms roots, zeros and x-intercepts, and to use the correct term in a given situation. The x-intercepts of the graph or the zeros of the quadratic function correspond to the roots of the quadratic equation. Students could be asked to find the roots of the equation x2 - 7x + 12 = 0, find the zeros of f (x) = x2 - 7x + 12, or determine the x-intercepts of y = x2 - 7x + 12. In each case they are solving x2 - 7x + 12 = 0 and arriving at the solution x = 3 or x = 4. 98 Discuss with students what it means to solve a quadratic graphically and the limitations of this method. They will then work with quadratics algebraically and decide when it is best to factor, complete the square or use the quadratic formula. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Journal • Ask students to write a paragraph explaining the relationship between the roots of a quadratic equation, the zeros of a quadratic function and the x-intercepts of the graph of a quadratic function. Pre-Calculus 11 (RF5.1) Paper and Pencil • Ask students to answer the following: (i) Find the zeros of f(x) = 2x2 + 5x - 7. (ii) Identify the x-intercepts of the graph. y 4.1 Graphical Solutions of Quadratic Equations 4 2 -6 -4 -2 2 4 6 x -2 -4 -6 Student Book (SB): pp.206-217 Teacher Resource (TR): pp.143148 BlackLine Master (BLM): 4-3, 4-4 -8 -10 (iii) Find the roots of 2x2 + 5x - 7= 0. (iv) What do you notice about the answers to the above questions? (RF5.1, RF5.2)) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 99 QUADRATIC EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF1 Factor polynomial expressions of the form: ax2 + bx + c, a ≠ 0 • a2x2 - b2y2, a ≠ 0, b ≠ 0 • a(f(x))2 + b(f(x)) + c, a ≠ 0 • a2(f(x))2 - b2(g(y))2, a ≠ 0, b ≠ 0 where a, b and c are rational numbers. [CN, ME, R] Students were first exposed to factoring polynomials in Mathematics 1201 (AN5). They removed the greatest common factor from the terms of a polynomial and factored polynomials of the form x2 + bx + c and ax2 + bx + c. As well, students factored perfect square trinomials, difference of squares and trinomials in two variables. In this unit, students will extend their knowledge of factoring trinomials and differences of squares to factoring polynomials of the form a2x2 - b2y2, a ≠ 0, b ≠ 0, a(f (x))2 + b(f (x)) + c, a ≠ 0 and a2(f (x))2 - b2(g (y))2, a ≠ 0, b ≠ 0. Achievement Indicators: RF1.1 Factor a given polynomial expression that requires the identification of common factors. RF1.2 Factor a given polynomial expression of the form: • ax2 + bx + c, a ≠ 0 • a2x2 - b2y2, a ≠ 0, b ≠ 0 RF1.3 Determine whether a given binomial is a factor for a given polynomial expression, and explain why or why not. RF1.4 Factor a given polynomial expression that has a quadratic pattern, including: • a(f (x))2+b(f(x))+c, a ≠ 0 • a2(f (x))2-b2(g(y))2, a ≠ 0, b ≠ 0 100 In Mathematics 1201, students worked extensively with factoring polynomial expressions. They were introduced to factoring using concrete and pictorial models and then moved to a symbolic representation. Most of the previous work dealt with integer coefficients. Remind students of the strategies used when factoring polynomial expressions, including the removal of the greatest common factor and the method of decomposition (AN5). In this course, students will be expected to be proficient with factoring at a symbolic level, including expressions with rational coefficients. Students should also be given the opportunity to apply their own personal strategies. Students should recognize when factoring expressions such as x2 + 6x + 8, possible binomial factors would only contain factors of 8. Therefore, a student should realize that x + 5, for example, would not be a possible factor of x2 + 6x + 8. This will improve their factoring skills which will be useful later when using factoring to solve a given quadratic equation. Factoring trinomials and differences of squares will be extended to factoring polynomials of the form a(f (x))2 + b(f (x)) + c, a ≠ 0 and a2(f (x))2 - b2(g (y))2, a ≠ 0, b ≠ 0. They can attempt these problems using different methods. Students can treat the expression as a single variable, continue to factor using their previous skills, and then use substitution to complete the factoring. Alternatively, they can expand and group like terms and then factor. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Performance • For the game Three in a Row, students are given a 5 × 5 game board which consists of 25 squares. Each square will have a quadratic equation written on it. Ask them to play in pairs and take turns solving each quadratic equation with the aim to be the first person to make three in a row. Pre-Calculus 11 (RF1.1, RF1.2) Paper and Pencil • Ask students to determine two values of n that will allow the polynomial 25b2 + nb + 49 to be a perfect square trinomial. They should use them both to factor the trinomial. (RF1.2) • Ask students to list possible binomial factors if the following expression x2 + bx + 24 can be factored. (RF1.3) 4.2 Factoring Quadratic Equations SB: pp.218-233 TR: pp.149-154 BLM: 4-3, 4-5 Interview • Ask students to respond to the following: (i) Explain why 9x2 − 16y2 can be factored but 9x2 + 16y2 cannot be factored. (RF1.2) (ii) Explain why (2x - 3y)2 ≠ 4x2 - 9y2. (RF1.2) • When Jenny factored x2 - x - 12, she said one of her binomial factors was x + 5. Without actually factoring the trinomial, ask students to explain why her response is incorrect. (RF1.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 101 QUADRATIC EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF1, RF5 Continued ... Achievement Indicators: RF1.4 Continued Consider the following example to compare the two methods: Factor: 2(x - 2)2 + 7(x - 2) + 5 Substitute p = x - 2 Expand (x - 2)2 2p2 + 7p + 5 2(x-2)(x-2)+7(x-2)+5 (2p + 5)(p + 1) 2(x2 - 4x + 4) + 7(x - 2) + 5 [2(x - 2) + 5][(x - 2 + 1] 2x2- 8x + 8 + 7x - 14 + 5 (2x - 4 + 5)(x - 2 + 1) 2x2 -1x - 1 (2x + 1)(x - 1) (2x + 1)(x - 1) Regardless of the method used, remind students that is is important to use brackets for all substitution. RF5.2 Continued Students solved a quadratic equation graphically by finding the x-intercepts. They will now solve using algebraic methods. They will explore different strategies to find values for the variable that make the function equal to zero. Students can use factoring to solve equations. Once a quadratic equation has been factored, they use the zero product property to determine the roots. If students were asked to solve 5x2 + 14x - 3 = 0, for example, 5x - 1 = 0 or x + 3 = 0. Remind students to substitute the value of each root into the original equation to verify that the value makes the equation true. Provide students with examples where a quadratic equation of the form ax2 + bx + c = 0 is missing the b or c value. Consider a quadratic equation where c = 0. When solving x2 - 5x = 0, for example, students might factor x(x - 5) = 0 and then solve by dividing both sides of the equation by x, resulting in only one solution. Ask students if this is correct and whether a root has been eliminated. Encourage students to check the reasonableness of their answers using the zero product property. Expose students to quadratic equations similar to x2 - 8 = 0 or (x - 2)2 - 49 = 0. They can isolate the squared term and take the square root of both sides of the equation. Remind students there are two possible solutions to these equations. They should also be asked to solve equations in which the right hand side of the equation does not equal 0 (i.e., 3x2 - 4 = x). 102 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Interview • Ask students to create an example of a quadratic function that cannot be factored. They should explain their reasoning. (RF1.1, RF1.2, RF1.3, RF1.4) • After factoring the different types of polynomials, ask students which ones they found easiest to factor and why. Which ones were most difficult and why? (RF1.1, RF1.2, RF1.3, RF1.4) Pre-Calculus 11 4.2 Factoring Quadratic Equations SB: pp.218-233 TR: pp.149-154 BLM: 4-3, 4-5 Paper and Pencil • Ask students to factor 4(x + 2)2 - 9(y + 1)2 using two methods. They should explain which method they prefer and why. (RF1.4) Performance • Two sets of different colored cards are required for this activity. One set will contain quadratic equations and the other set will have their corresponding solutions. Ask students to lay out the cards and match the equation card with its corresponding solution card. x 2 − 2 x − 24 = 0 x = 5 ; x = 2 2 2 x 2 + 5x + 3 = 0 3 x = ; x = −1 2 2 x 2 − 5x + 3 = 0 2x2 + 7x + 5 = 0 3 x = − ; x =1 2 x = 4 ; x = −6 x 2 + 2 x − 24 = 0 2 x 2 − 9 x + 10 = 0 2x2 − x − 3 = 0 x = −4 ; x = 6 2x2 + x − 3 = 0 5 x = − ; x = −1 2 3 x = ;x =1 2 3 x = − ; x = −1 2 Web Link The following is an interactive site consisting of a quadratic solver and its graphical representation. www.mathisfun.com/quadraticequation-solver.html (RF5.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 103 QUADRATIC EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF5 Continued ... Achievement Indicators: RF5.2 Continued RF5.3 Derive the quadratic formula, using deductive reasoning. When students are exposed to quadratic equations that cannot be factored, they have the option to complete the square or apply the quadratic formula. The process of completing the square was introduced in the previous unit when students converted a quadratic function from standard form to vertex form. They will now use completing the square to determine the roots of a quadratic equation in standard form. Once students solve a quadratic equation using this method, provide students the graph of the corresponding function to reinforce the connection between the roots of the equation and the x-intercepts of the graph. It is important for students to understand how the quadratic formula is developed before they apply it to quadratic equations in standard form. They should use a numerical example before moving to the general form ax2 + bx + c = 0. Ask them to complete the square using an example similar to 3x2 - 7x + 1 = 0. Assist students as they follow the same procedure to derive the quadratic formula for ax2 + bx + c = 0. Once the quadratic formula has been derived it can be used to find the roots of any quadratic equation in standard form. 3x 2 − 7x + 1 = 0 3( x 2 − 73 x ) + 1 = 0 49 3( x 2 − 73 x + 49 ) − 12 +1= 0 36 ax 2 + bx + c = 0 a( x 2 + ba x ) + c = 0 a( x 2 + ba x + b2 4a2 )− ab 2 4a 2 +c =0 ( x − 76 )2 = 37 36 b )2 − b + c = 0 2a 4a 2 2 a( x + 2ba ) − 4b a + 44aca = 0 2 a( x + 2ba )2 = b 4−a4 ac 2 ( x + 2ba )2 = b −42ac 4a x − 76 = ± 37 6 x+ b 2a =± b 2 −4 ac 4a2 x+ b 2a =± b 2 −4 ac 2a x = − 2ba ± b 2 −4 ac 2a 49 3( x − 76 )2 = 12 −1 49 − 12 3( x − 76 )2 = 12 12 37 3( x − 76 )2 = 12 x = 76 ± 37 6 x = 7± 6 37 2 a( x + x= −b ± b 2 −4 ac 2a 2 Students may need assistance when adding rationals ( − 4b a + c ) and simplifying variable roots ( 4a 2 ). These topics are covered in greater detail later in this course in the Radical Expressions Unit and the Rational Expressions Unit. 104 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Performance • For the game Quadratic Bingo, a list of approximately 50 possible sets of solutions to various quadratic equations will be placed on the board. Each student is given a blank bingo card. They will randomly fill in the 24 squares of their card with one set of solutions. They are now ready to play BINGO. Present one quadratic equation at a time for students to solve. If the solutions are present on their card, the square is covered. Continue to provide equations to be solved until a student has completed a diagonal, horizontal or vertical line on the card. Pre-Calculus 11 4.3 Solving Quadratic Equations by Completing the Square SB: pp.234-243 TR: pp.155-159 BLM: 4-3, 4-6 4.4 The Quadratic Formula BINGO SB: pp.244-256 TR: pp.160-165 BLM: 4-3, 4-7 FREE Web Link (RF5.2) www.k12pl.nl.ca/seniorhigh/ introduction/math2201/ classroomclips.html The Quadratic Formula clip demonstrates students taking turns solving a quadratic equation using the quadratic formula. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 105 QUADRATIC EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF5 Continued... Achievement Indicators: RF5.4 Identify and correct errors in a solution to a quadratic equation. It is beneficial to have students analyze solutions that contain errors. Students should be provided with worked solutions of quadratic equations that may or may not contain errors. If errors are present, students should identify the error and provide the correct solution including why/how the error occurred. This reinforces the importance of recording solution steps rather than only giving a final answer. Some common errors occur when students are simplifying the quadratic formula. These include: • Apply the quadratic formula without ensuring the equation is written in standard form. • Use x = −b ± b 2−a4 ac rather than the correct form of the quadratic formula. • Incorrectly produce two possible common errors if the b value is negative. 2 (i) If b = -2 then -b = -(-2) = -2 (ii) If b = -2 then b2 = -22 = -4 • Incorrectly simplify when applying the quadratic formula. (i) 8± 5 2 =4± 5 (ii) 2 + 42 5 = ± 2 5 • Do not recognize that the ± results in two solutions. Suggest that students work through the solutions separately, showing calculations for both the positive solution and the negative solution. RF5.5 Select a method for solving a quadratic equation, justify the choice, and verify the solution. As students work through the various strategies for solving quadratic equations, they should realize that sometimes one method is more efficient than another. The method students choose to solve a quadratic equation will depend on the way the equation is presented. RF5.6 Explain, using examples, how the discriminant may be used to determine whether a quadratic equation has two, one, or no real roots; and relate the number of zeros to the graph of the corresponding quadratic function. Students have solved quadratic equations graphically and algebraically. Regardless of their strategy to solve the equation, they should be able to connect the fact that their algebraic solution is the same as the x-intercepts of the graph. In the previous unit, when functions were written in the form y = a(x - p)2 + q , students investigated how the values of a and q affected the number of x-intercepts. They will now explore how these situations arise when solving equations using the quadratic formula. 106 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to identify the most efficient strategy (i.e., determining square roots, factoring, completing the square, applying the quadratic formula or graphing its corresponding function) when solving each equation. They should justify their choice. (i) (ii) (iii) (iv) (v) (x + 2)2 - 1 = 15 49x2 - 64 = 0 3x2 - 11x + 6 = 0 x2 - 10x - 24 = 0 3x2 + 8x + 7 = 0 4.4 The Quadratic Formula SB: pp.244-256 TR: pp.160-165 BLM: 4-3, 4-7 (RF5.5) • Given the quadratic equation ax + bx + c = 0, ask students what relationship must be true for the coefficients a, b, and c so that the equation has: 2 (i) two distinct real roots (ii) two equal real roots (iii) no real roots (RF5.6) Observation • Mary, David, and Ron are students in a group. They are given the equation A = x2 + 3x – 110 where A represents the area of a field and x represents the width in metres. The students were asked to find the width if the area was 100 m2. Each student decided to solve the equation using their own preferred method. Here are their solutions: Mary 2 Ron x + 3x − 110 = 100 x 2 + 3x − 110 = 0 x 2 + 3x − 210 = 0 x 2 + 3x − 10 = 0 (x + 5)(x − 3) = 0 x= x= 3 ± 9 − (4)(1)( −210) 2 3 ± −831 x= 2 3 ± 28.827 x= 2 x = 15.9 or x = −12.9 width is 15.9 m David x + 3x − 110 = 100 2 x = −5 or x = 3 width is 3 m −3 ± 9 − (4)(1)( −110) 2 −3 ± 9 + 440 x= 2 −3 ± 449 x= 2 −3 ± 21.2 x= 2 x = 9.1 or x = −12.1 width is 9.1 m Ask students to identify and explain any errors in the students’ work. They should then proceed to write the correct solution. (RF5.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 107 QUADRATIC EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF5 Continued... Achievement Indicators: RF5.6 Continued When students use the quadratic formula to find roots of a quadratic equation they will be exposed to trying to find the square roots of nonperfect squares and negative numbers. It is important for students to distinguish between exact solutions and approximate solutions. Although, it is not the intention of this course to introduce students to the imaginary number system, they need to be aware that there is no real solution to the square root of a negative number. This will lead to a discussion regarding the conditions that are necessary for the quadratic formula to result in two real roots, one real root and no real solutions. Ask students to solve each of the following using the quadratic formula. A: 2x2 + 10x + 3 = 0 B: x2 + 6x + 9 = 0 C: 2x2 + 3x + 5 = 0 As teachers observe students’ work, use the following prompts to promote discussion: RF5.7 Solve a problem by: • analyzing a quadratic equation • determining and analyzing a quadratic equation. • What is the value under the square root in each equation? What does this tell you about the roots of the equation? • What connection can be made between the value of b2 - 4ac and the number of real roots an equation has? • What values of b2 - 4ac could lead to approximate answers? Quadratic equations can be used to model a variety of situations such as projectile motion and geometry based word problems. Students should be exposed to examples which require them to model the problem using a quadratic equation, solve the equation and interpret the solution. Consider the example: • A rectangular lawn measuring 8 m by 4 m is surrounded by a flower bed of uniform width. The combined area of the lawn and flower bed is 165 m2. What is the width of the flower bed? It is important for students to recognize that the context of the problem dictates inadmissible roots. Discuss with students different scenerios that produce inadmissible roots. For example, time, height and length, would not make sense if they have a negative numerical value. However, temperature could be both negative and positive. 108 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 When Chantal was asked to describe the roots of the equation 14x2 – 5x = – 5, she rearranged the equation so that it would equal zero, then used the quadratic formula to find the roots. Her workings are shown below. Edward said that she didn’t have to do all that and he then showed the class his work. Ask students if they are both correct. They should identify their preferred method and explain their reasoning. Chantal Edward 5 ± 25 − 280 28 5 ± −255 x= 28 x= no real roots • 4.4 The Quadratic Formula SB: pp.244-256 TR: pp.160-165 BLM: 4-3, 4-7 b 2 − 4ac = 25 − 280 = −255 no real roots (RF5.6) Ask students to answer the following: (i) For what values of t does x2 + tx + t + 3 = 0 have one real root? (RF5.6) (ii) Show that if the quadratic equation px2 +(2p + 1)x + p = 0 has two real unequal roots, then 4p + 1 > 0. (RF5.6) (iii) Assume a, b and c are real numbers. How many times would y = ax2 + bx + c, intersect the x-axis if the discriminant of ax2 + bx + c = 0 is (a) positive? (b) zero? (c) negative? (RF5.6) (iv) Create quadratic equations that have two distinct roots, two equal roots, and no real roots. They should explain their reasoning. (RF5.6) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 109 QUADRATIC EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF5 Continued... Achievement Indicator: RF5.7 Continued When modelling situations for students, emphasize that restrictions sometimes need to be placed on the independent variable of the function. If a solution does not lie in the restricted domain, then it is not a solution to the problem. The following is an example with a restricted domain: • A baseball is thrown from an initial height of 3 m and reaches a maximum height of 8 m, 2 seconds after it is thrown. At what time does the ball hit the ground? In the above example, the quadratic equation only models the path of the ball from the time it leaves the throwers’ hand to the time it makes first contact with the ground. This quadratic equation yields two possible solutions, one of which is negative. This implies that it occurred before the ball was thrown. The restriction on the domain causes the negative solution to be inadmissible since time cannot be negative and only the positive solution is accepted. 110 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 QUADRATIC EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • A ball is thrown from a building at an initial height of 11 metres and reaches a maximum height of 36 metres, 5 seconds after it is thrown. Ask students to do the following: Pre-Calculus 11 (i) Write a quadratic equation which models this situation. (ii) Three targets are placed at different locations on the ground. One is at (10,0), another at (11,0) and a final target is placed at (12, 0). Which target does the ball hit? Explain how you arrived at your answer. (RF5.7) SB: pp.244-256 TR: pp.160-165 BLM: 4-3, 4-7 • 4.4 The Quadratic Formula Ask students to find two consecutive whole numbers such that the sum of their squares is 265. (RF5.7) • A diver’s path when diving off a platform is given by d = -5t2 + 10t + 20, where d is the distance above the water (in feet) and t is the time from the beginning of the dive (in seconds). (i) How high is the diving platform? (ii) When is the diver 25 feet above the water? (iii) When does the diver enter the water? (RF5.7) • Ask students to choose a quadratic word problem from the class notes or group workstations and use it as a guide to create their own word problem (encourage them to use a real-life situation that they are interested in). Remember to have them include their solutions on a separate sheet. Students could give their problem to another student to solve. (RF5.7) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 111 QUADRATIC EQUATIONS 112 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Radical Expressions and Equations Suggested Time: 13 Hours RADICAL EXPRESSIONS AND EQUATIONS Unit Overview Focus and Context In this unit, students will simplify radical expressions and perform the four operations on these expressions (addition, subtraction, multiplication and division). When working with division, students will be expected to rationalize denominators containing radicals. They will also identify the restrictions on the values of variables in radical expressions that are real numbers. Students will solve problems that involve radical equations, limited to square roots, and identify the extraneous roots. Outcomes Framework GCO Develop algebraic reasoning and number sense. SCO AN2 Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands. SCO AN3 Solve problems that involve radical equations (limited to square roots). 114 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Mathematics 2200 Mathematics 3200 Algebra and Number Algebra and Number Relations and Functions AN2 Demonstrate an understanding of irrational numbers by: • representing, identifying and simplifying irrational numbers AN2 Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands. RF13 Graph and analyze radical functions (limited to functions involving one radical). [CN, R, T,V] [CN, ME, PS, R] • ordering irrational numbers. AN3 Solve problems that involve [CN, ME, R, V] radical equations (limited to square roots). [C, PS, R] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 115 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN2 Solve problems that involve operations on radicals and radical expressions with numerical and variable radicands. [CN, ME, PS, R] In Grade 9, students determined the square root of a perfect square and worked with benchmarks to approximate the square root of non-perfect square rational numbers (9N5, 9N6). In Mathematics 1201, students worked with mixed and entire radicals, limited to numerical radicands (AN2). The notation n x was introduced, where the index of the radical was a maximum of 5. In this unit, students will simplify radical expressions with numerical and variable radicands and will add, subtract, multiply and divide these expressions. When working with division, students will be responsible for a monomial and binominal denominator. They will also identify values of the variable for which the radical expression is defined. Students will apply and use radicals to solve equations involving radical expressions. Solutions will be verified by substitution and extraneous roots will be explored. Achievement Indicators: AN2.1 Compare and order radical expressions with numerical radicands in a given set. AN2.2 Express an entire radical with a numerical radicand as a mixed radical. AN2.3 Express a mixed radical with a numerical radicand as an entire radical. In Mathematics 1201, students expressed a radical as either a mixed or entire radical with numerical radicands. Review this concept with students and reinforce that if radicals have the same index, the radicands can be compared. It is helpful to rearrange the mixed radical as an entire radical for the purpose of ordering and estimation without the use of technology. When given two right triangles, for example, ask students to determine which hypotenuse length is greater, 3 5 or 4 3 . Although students could use a calculator to approximate the length, the focus here is to rewrite the numerical radicals in equivalent forms and make a comparison. Part of this unit is a review of topics taught in Mathematics 1201, with the exception of the introduction to secondary roots. Refer to the Roots and Powers unit in the Mathematics 1201 Curriculum Guide for further information regarding teaching strategies and common student errors (AN2.6, AN2.7, AN2.8). Students should have some exposure to principal and secondary square roots. They should recognize that every positive number has two roots. For example, the square root of 49 is 7 since 72 = 49. Likewise (-7)2 = 49 so -7 is also a square root of 49. The value 49 = 7 is called the principal square root and − 49 = −7 is the secondary square root. Although students will compare principal and secondary square roots, it is equally important for them to understand why it makes sense to only use the principal square root in certain situations. For example, if students are using the Pythagorean theorem to calculate the length of a leg of a right triangle, length is a positive number, so the square root must also be positive. 116 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Set up a clothesline across the whiteboard to represent a number line with several benchmarks identified. Each student is given a card with an expression of mixed or entire radical. Ask them to pin the card along the number line. They should be able to explain why they placed the card in that position. • • (AN2.1, AN2.2 AN2.3) Each group (3-4 students) will be given a deck of cards. Each card will have a different mixed radical. The group will then work together to sort the cards from largest to smallest. The first group with the cards sorted in the correct order wins the competition. (AN2.1, AN2.2 AN2.3) Students can play the Radical Matching Game in groups of two. Give students a deck of cards containing pairs that display equivalent mixed radicals and entire radicals. All cards should be placed face down on the table. The first student turns over 2 of the cards, looking for a pair. If they get a pair, they remove the cards and go again. If the overturned cards do not form a pair, it is the other player’s turn. The player with the most matches at the end of the game wins. 5.1 Working with Radicals Student Book (SB): pp.272-281 Teacher Resource (TR): pp.179185 Blackline Master (BLM): 5-3, 5-4 (AN2.1, AN2.2 AN2.3) Web Link The information regarding roots and powers from the Mathematics 1201 Curriculum Guide can be found at: www.ed.gov.nl.ca/edu/k12/ curriculum/guides/index.html ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 117 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN2 Continued ... Achievement Indicator: AN2.4 Explain, using examples, that (-x)2 = x2, x 2 = x and 2 x ≠ ± x ; e.g., 9 ≠ ±3 . When asked to simplify x 2 , students may initially conclude that 2 x = x . Prompt student discussion using the following questions: • Does this happen with all positive values of x? • Does this happen with all negative values of x? Ask students to simplify ( −5)2 . The following is a sample of student answers: 2 2 1 ( −5) = ( −5) 2 = ( −5) = −5 2 ( −5) = 25 = 5 This is an opportunity for discussion around the correct answer when the value of x is negative. Some students may challenge the incorrect solution provided in column one above. It is important for students to m recognize that when a is a negative number, a n is not defined because it is not possible to define such expressions consistently. Students can 1 compare this value when n is even to when it is odd. (i.e., ( −4) 2 = −4 is undefined under the set of real numbers but ( −8) = 3 −8 is defined). 1 3 This is a great opportunity for discussion as to what is the correct answer when the value of x is negative. Although it is common for students to replace x 2 with x, students must recognize that this is correct when x ≥ 0. They should also realize that x 2 is equivalent to -x when x ≤ 0. Consider the following: When x = 4: 2 (4 ) 2 16 x = x = 2 2 ( −4 ) 2 16 x = x = 2 2 2 x =4 x =4 ∴ x 2 When x = -4: =x 2 x = −x ≠x Although students will be exposed to the concept of absolute value later in this course, an introduction to absolute value is important here. Ensure students understand the absolute value symbol produces a result that is always positive, which is the principal square root. Hence x 2 = x . Once this idea has been established, students can assume, for the sake of simplicity, all variables are positive. Therefore x 2 = x . ∴ x 118 2 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Authorized Resource Journal • Pre-Calculus 11 Ask students to respond to the following: 2 (i) Your friend stated that since 32 = 9 and (-3)2 = 9, then x is always ±x. Use specific examples to show whether you agree or disagree with his statement. (AN2.4) 1 5.1 Working with Radicals SB: pp.272-281 TR: pp.179-185 BLM: 5-3, 5-4 1 (ii) Explain why ( −16) 2 is undefined, where as ( −64) 3 is defined under the set of real numbers. (AN2.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 119 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN2 Continued ... Achievement Indicators: AN2.5 Identify the values of the variable for which a given radical expression is defined. In Mathematics 1201, students simplified radicals with numerical radicands. They also determined the domain of a variety of functions (RF1). This is their first exposure to simplifying expressions with variable radicands. They will write the restrictions on the variable and then write the expression in its simplest form. Students should be comfortable recognizing square roots of nonnegative numbers are defined under the set of real numbers. Remind students of this concept using any number of numerical examples such as 4 = 2 while −4 is not defined. Expose students to examples similar to 16 , 4 −81, 3 −27 , 3 8 , 6 −64 . Students should recognize that if a radical has an even index, the radicand must be non-negative. If a radical has an odd index, the radicand can be any real number, including negative numbers. This could lead into a discussion of what happens if the radicand is variable in nature such as x . Reinforce the concept that the domain of a square root function is limited to values for which the function has meaning. Use examples to allow students to intuitively investigate variable expressions as radicands and then progress to actually solving an inequality algebraically. While students solved inequalities in Grade 9(9PR4), it may be necessary to review the various rules used to solve inequalities. AN2.6 Express an entire radical with a variable radicand as a mixed radical. AN2.7 Express a mixed radical with a variable radicand as a entire radical. 120 Students will convert an entire radical with one variable to a mixed radical and will then progress to multiple variables. They will also work backwards and express a mixed radical with a variable as an entire radical. When simplifying 4 x 7 , for example, students can rewrite as 4 4 4 3 3 x x = x x . For the radical to represent a real number, x ≥ 0 because the index is a real number. When writing 2 x 3 5 x 2 as an entire radical, the variable x in the expression 3 2 3 x 3 5x 2 can be any real number since the index of the radical is an odd number. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Observation Authorized Resource • Pre-Calculus 11 x Ask students to complete the following chart and make predictions about the restrictions on each variable. x 1 x 1 x x −1 x +1 1 x −1 1 x +1 -3 -2 -1 0 1 2 3 5.1 Working with Radicals SB: pp.272-281 TR: pp.179-185 BLM: 5-3, 5-4 As teachers observe students’ work, use the following questions to guide discussion: (i) What values of x were undefined? What values of x were defined? (ii) Is the restriction different if the radical expression is in the denominator? (iii) How could solving inequalities help when determining the restriction? (AN2.5) Performance • For the activity Sticky Bars, present students with a selected response question where they could be expected to convert an entire radical with a variable radicand to a mixed radical, or vice versa. The answer is anonymously recorded on a post it note and submitted to the teacher. The teacher or student volunteer arranges the sticky notes on the wall or whiteboard as a bar graph representing the different student responses. Have a discussion regarding why students may have selected the answers they did. (AN2.6, AN2.7) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 121 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN2 Continued... Achievement Indicator: AN2.8 Perform one or more operations to simplify radical expressions with numerical or variable radicands. Students will add and subtract radicals that contain numerical and variable radicands before they move into multiplication and division. This could be introduced by asking students to add 2x + 3x. Use leading questions such as the following: • What process is involved in the addition of monomials? • How is this expression similar to 2 7 + 3 7 ? • What are like radicals? • How can this process be applied to radicals? The goal is for students to realize adding and subtracting radical expressions is comparable to combining variable expressions with like terms. It is necessary for the radical to have the same index and the same radicand. Students can then apply the same strategies with indices other than 2. The distributive property can also be applied when simplifying sums and differences of radical expressions. Ask students how to rewrite 2x + 3x in another form, namely (2 + 3)x. Similarly, 2 7 + 3 7 can be expressed as (2 + 3) 7 = 5 7 . Check students’ understanding by asking them to express −4 3 as the sum of two like radicals. It is important for students to recognize that, even when adding or subtracting like radicals, the solution can require further simplifying. When subtracting the expression 3 8 − 7 8 , for example, the solution −4 8 can be simplified to −8 2 . It would be good practice to simplify the radical before like terms are combined. This is especially beneficial when working with large numerical radicands. Students should also be exposed to examples, such as, 2 18 + 3 50 − 5 2 or 4 3 54 + 3 −45 − 3 2 , where it is necessary to simplify one or more radicals in order to complete the addition and/or subtraction operations. Common errors occur when adding or subtracting radicals. When asked to add 4 + 2 3 , for example, they may write 6 3 . Students should be encouraged to check their answers by expressing the sum/difference using the distributive property. Another common error occurs when students incorrectly apply the operations of addition and subtraction to the radicands. The value of 31 3 + 17 3 , for example, does not equal 48 6 . This error may occur more often once students have been introduced to the multiplication of radicals. 122 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 The voltage V required for a circuit is given by V = PR where P is the power in watts and R is the resistance in ohms. How many more volts are needed to light a 100-W bulb than a 75-W bulb if the resistance for both is 100 ohms? They should solve the problem in exact and approximate form. (AN2.8) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 5.1 Working with Radicals SB: pp.272-281 TR: pp.179-185 BLM: 5-3, 5-4 123 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN2 Continued... Achievement Indicator: AN2.8 Continued Student learning will continue with addition and subtraction involving variable radicands. Ask students the following questions: • How is adding and subtracting algebraic expressions with radicals similar to adding and subtracting numerical expressions with radical values? Use examples similar to 8 x 3 − 4 2 x where x ≥ 0, and −2 3 16x 4 + 5x 3 54 x and 2 45 − 6 20 to explain your reasoning. • What strategy did you use? Similar to adding and subtracting radicals, students will multiply and divide radicals beginning with numerical radicands. To demonstrate the multiplication property of radicals, reiterate the relationship between a radical and a power with rational exponents. 1 1 1 1 2 3 2 × 5 2 = (3 × 5) 2 = 15 ⇔ 3 × 5 = 3 × 5 = 15 In Mathematics 1201, students applied the laws of exponents to rational exponents (AN3). Encourage students to look for a pattern through the use of several examples. The multiplicative property of radicals can then be introduced to students, a × b = ab where a ≥ 0 and b ≥ 0. This property can also be used to discuss why two radicals with the same index can be multiplied. Although students were exposed to this property in Mathematics 1201, it was used exclusively for the purpose of expressing a radical as a mixed or entire radical (AN2). Students should be given an opportunity to further explore the product rule for radicals (i.e., c a × d b = cd ab ) and the commutative property of multiplication. Ask them to rewrite the expression 1 1 2(3 2 ) × 5(6 2 ) using radicals to generalize a pattern. Students’ workings may vary but the result should be the same. Consider the following sample: 1 1 2(3 2 ) × 5(6 2 ) 2( 3 ) × 5( 6 ) 1 1 (2 × 5)(3 2 × 6 2 ) 1 (10)(3 × 6) 2 10(18) 1 2 (2 × 5) (10) ( ( 3× 6 3×6 ) ) 10 18 Students will also be exposed to examples where the index is not 2. They should be introduced to the rule c n a × d n b = cd n ab , where n is a natural number and c, d, a, and b are real numbers. They should recoginze that if n is even then a ≥ 0 and b ≥ 0. 124 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Authorized Resource Performance • Create centres in the classroom containing worked solutions of simplifying radical expressions with numerical or variable radicands. Students will participate in a carousel activity where they are asked to move throughout the centres to identify and correct errors. Samples are shown below: Pre-Calculus 11 5.1 Working with Radicals SB: pp.272-281 TR: pp.179-185 BLM: 5-3, 5-4 25 5 + 13 5 (i) = 38 10 (ii) 3 18 x + 2 8 x 3 = 3 2x + 4 = 7 4x = 14 x 5.2 Multiplying and Dividing Radical Expressions 3 2x 3 SB: pp.282-293 TR: pp.186-193 BLM: 5-3, 5-5 6 3 (AN2.8) Paper and Pencil • Ask students to simplify each of the following: (i) 2 18 + 9 7 − 63 3 3 3 (ii) 6 32 x − 5 8 x + 3 2 x 3 4 (iii) −5 3 256x + 192 x (AN2.8) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 125 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN2 Continued... Achievement Indicator: AN2.8 Continued Advise students that they are less likely to make simplification errors if they simplify radicals before multiplying. An example such as the following could be used to illustrate the two methods: Multiply first: 80 × 12 = 960 = 8 15 Simplify first: 80 × 12 = 4 5 × 2 3 = 8 15 Ask students which method they prefer and why. Students will also multiply radicals using the distributive property. It may be helpful to walk them through the similarities between multiplying radical expressions and multiplying polynomials. In order to avoid common errors, this would be a good opportunity to reinforce the commutative and associative property of multiplication. When multiplying 5 × 3 , for example, students may write 15 . They may understand their error if they apply the commutative property to rewrite the expression as 3 5 . Another error occurs when students are asked to multiply an expression such as 3 5 × 6 and their result is 90 . This can be avoided if the associative property is used to re-order the expression as 3 5 × 6 = 3 30 . ( ) Students will explore how multiplying and dividing algebraic expressions with radicals is similar to multiplying and dividing numerical expressions with radical values. Ask them to use examples similar to −4 12 × −2 18 and ( −4 x ) × ( −2 x 2 ) where x ≥ 0, and 3 2 ( −4 3 x ) × ( −2 x ) to explain their reasoning. 126 • What strategy did you use? • When is it necessary to use the distributive property to multiply expressions that contain radicals? Create an example and show the solution. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to simplify the following: (i) ( 2 + 6) 5.2 Multiplying and Dividing Radical Expressions 2 (ii) (3 8 − 4 )(2 + 7 3 ) (iii) ( 20 + SB: pp.282-293 TR: pp.186-193 BLM: 5-3, 5-5 24 )(3 12 − 5 32 ) (iv) -2 3 12(4 3 2 − 5 3 9 ) (v) ( −3 x )(6 x 3 ) ( vi) (3 x + 2)(3 − 5 x ) (AN2.8) Performance • Ask students to participate in Commit and Toss. Provide students with a selected response problem as shown below. They anonymously commit to an answer and provide a justification for the answer they selected. Students crumble their solutions into a ball and toss the papers into a basket. Once all papers are in the basket, ask students to reach in and take one out. Ask students to then move to the corner of the room designated to match their selected response. In their respective corners, they should discuss the similarities or differences in the explanations provided and report back to the class. Web Link www.k12pl.nl.ca/seniorhigh/ introduction/math2201/ classroomclips.html The Radical Expressions clip demonstrates students performing operations to simplify a radical expression using the activity Commit and Toss. Express ( 3 − 2 )2 in simplest form. (A) 1 (B) 5 − 2 6 (C) 1 − 2 3 (D) 5 Explain your reasoning: (AN2.8) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 127 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN2 Continued... Achievement Indicator: AN2.9 Rationalize the denominator of a rational expression with monomial or binomial denominators. Division of radicals and the rationalization of the denominator are new concepts for students. The denominator can be a monomial or a binomial. The rules of exponents should be integrated when introducing the division of radicals. For example: ( ) 4 9 1 2 1 2 = 41 ⇔ 92 4 9 ( ) 8 27 = 4 9 1 3 1 3 8 = 38 = 8 1 ⇔ 3 27 3 27 27 3 Ask students to predict the rule when dividing radicals. Using the examples, reinforce to students that they can only divide radicals that have the same index. They should recognize the quotient rule of radicals states that the nth root of a quotient is the quotient of the nth root. In n other words, n ba = n ba where n is a natural number, a and b are real numbers, and b ≠ 0. Remind students if n is even then a ≥ 0 and b > 0. Students should be exposed to a variety of cases when simplifying radicals with fractions. Include examples such as the following: 25 25 5 = = 4 2 4 12 6 = 12 6 = 2 12 5 = 12 5 = 2 3 5 As students simplify radicals such as these, they should ask themselves the following questions: • Is the denominator a perfect root? • Can the numerator and denominator divide into a rational number? • Will the denominator have a radical when simplified? Students will develop a strategy for converting a fraction that has radicals in its denominator into an equivalent fraction with no radicals in the denominator. Rationalizing the denominator provides a standard notation for expressing results. Using an example such as 2 3 , ask them 5 what they can multiply the numerator and denominator by that results in a rational expression. 128 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Authorized Resource Performance • In groups of two, ask students to participate in the activity Pass the Problem. Give each pair a simplification problem that involves rationalizing the denominator. Ask one student to write the first line of the solution and pass it to the second student. The second student will verify the workings and check for errors. If there is an error present, ask students to discuss the error and why it occurred. The student will then write the second line of the solution and pass it their partner. This process continues until the solution is complete. Pre-Calculus 11 5.2 Multiplying and Dividing Radical Expressions SB: pp.282-293 TR: pp.186-193 BLM: 5-3, 5-5 (AN2.9) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 129 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN2 Continued... Achievement Indicator: AN2.9 Continued Expose students to examples where the index is greater than 2. For 3 2 example, when students simplify 3 5 , they may initially multiply the numerator and denominator by the term 3 5 . They should recognize a radical still remains in the denominator. When rationalizing the denominator, the root should guide student choices. They multiply numerator and denominator by a value where the powers of the 2 3 denominator will equal the index of the root. Therefore, 2 ⎛⎜ ( 5 ⎞⎟ 3 1 5 ⎝ ( 52 ⎠ results in a rational denominator. 3 3 This strategy can be applied to variable monomial denominators. Given 4 an example, such as 4 7 x2 2 , the index will help students determine 27 x y what expression they should use to rationalize the denominator. In other words, 4 7 x ⎛ 3 x y ⎞ will produce a rational denominator. ⎟ 4 33 x 2 y 2 ⎜ ⎝ 3x y ⎠ 4 1 2 2 4 1 2 2 2 3 Examples such as 7 5 , where the denominator is a mixed radical should be included. Encourage students to think about what they would have to multiply by to rationalize the denominator. The initial tendency may be to multiply the numerator and denominator by 7 5 . Although it is not the most efficient strategy, it is correct. However, the resulting expression will have to be simplified. 5 They should realize multiplying by 5 will rationalize the denominator. Students should first be exposed to expressions with entire radicals in both the numerator and denominator. This should then be extended to include mixed radicals and examples where there is more than one term in the numerator. It is the students’ choice whether they simplify before or after they rationalize the denominator. When working with larger numbers, however, simplifying first would allow them to work with smaller numbers. AN2.10 Describe the relationship between rationalizing a binomial denominator of a rational expression and the product of the factors of a difference of squares expression. 130 1 Students will simplify an expression such as 2 + 3 where it is necessary to rationalize a binomial denominator. Initially, they may think they can multiply the numerator and denominator by 3 , as in their previous work with monomial denominators. As they explore this, however, they should discover that 2 +1 3 × 33 = 2 33+ 3 , and the denominaor still contains a radical. Prompt them to multiply the numerator and denominator by the conjugate of 2 + 3 . This is similar to multiplying the factors of a difference of squares expression. Students should also be exposed to expressions where both terms in the binomial are irrational. Expressions involving variable numerators and denominators should be explored in a similar fashion. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil Pre-Calculus 11 • Ask students to rationalize the denominator for each of the expressions: (i) (ii) (iii) (iv) 5.2 Multiplying and Dividing Radical Expressions 7 1+ SB: pp.282-293 TR: pp.186-193 BLM: 5-3, 5-5 7 2+3 5 2 5 −4 b a+ b x +3 y y −x (AN2.9, AN2.10) Journal • When asked to rationalize the denominator in the expression 4 2 + 7 your friend said he could just multiply the expression by Ask students to explain why this would not work. 7 7. (AN2.9, AN2.10) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 131 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN3 Solve problems that involve radical equations (limited to square roots). [C, PS, R] Achievement Indicators: Students will solve radical equations involving square roots. It is intended that the equations will have no more than two radicals and the radicand will contain variables that are first and second degree. Students will be responsible for solving equations resulting in a linear or quadratic equation. AN3.1 Determine any restrictions on values for the variable in a radical equation. Earlier in this unit, students determined restrictions within the real number system on a variable in a radical that had an even index. They will continue to determine the restrictions before solving a radical equation. AN3.2 Determine the roots of a radical equation algebraically, and explain the process used to solve the equation. This would be a good opportunity for students to compare a radical equation to its graph to develop an understanding of restrictions for the variable and the points that satisfy the equation. Using a table of values, ask students to graph y = x . . y AN3.3 Verify, by substitution, that the values determined in solving a radical equation algebraically are roots of the equation. AN3.4 Explain why some roots determined in solving a radical equation algebraically are extraneous. 4 (4,2) 2 -4 -2 2 -2 x 4 -4 Focusing on the point (4,2), ask students how they would algebraically solve the equation given only the y-coordinate 2. When solving 2 = x , the value of x can be determined by inspection. Students could also use the idea that squaring a number is the inverse operation of taking the square root. This technique may seem straight forward but students should be exposed to equations where the value is not a solution to the original equation. Consider the example 2 x − 1 = −3 . The left-hand side of the equation calls for a positive square root, but the right-hand side of the equation is negative. Intuitively, there can be no solution. y 4 y = 2x – 1 2 -4 -2 2 -2 -4 132 4 x y = -3 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Ask students to explain why the domain of 2 x − 5 is x ≥ 52 , while 1 the domain of 2 x −5 is x > 52 . They should illustrate their answers graphically with technology and then algebraically. Pre-Calculus 11 • (AN3.1) Ask students to state the restrictions, solve and check for extraneous roots. (i) 4x = 8 (ii) x +4 =5 5.3 Radical Equations SB: pp.294-303 TR: pp.194-199 BLM: 5-3, 5-6 (iii) 2 x − 3 = −2 (AN3.1, AN3.2, AN3.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 133 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN3 Continued... Achievement Indicators: AN3.1, AN3.2 AN3.3, AN3.4 Continued Students should recognize that given the y-coordinate of -3, there is no possible x-coordinate that satisfies the equation. However, squaring both sides of the equation results in x = 5. This cannot be correct, as both substitution and the graph have shown that the equation has no solution. This is a great lead in to the concept of extraneous roots. Extraneous roots occur because squaring both sides and solving the equation may result in roots that do not satisfy the original equation. As students solve equations, reinforce the importance of checking that the value is a solution to the original equation. Any extraneous roots are rejected as answers. Another strategy students can use to solve a radical equation involves applying a power that will eliminate the radical expression on both sides of the equation. To solve x + 1 = 4 , for example, students would first rewrite the radical expression with a rational exponent, resulting in 1 2 ( x + 1) 2 = 4. Using the multiplicative inverse gives ( x + 1) = (4)2 and leads to an equation without radicals. ( 1 2 ) Students should be exposed to equations where the radical is not isolated. In such situations, make a comparison to solving a linear equation. They should recognize that solving an equation such as 3 + 2 x + 1 = 7 follows a process that is similar to solving 3 + x = 7. Rather than squaring both sides of a radical equation, students sometimes mistakenly square the individual terms. When solving 3 + 2 x + 1 = 7, for example, they may not isolate the radical. Squaring each term results in the incorrect equation 32 + ( 2 x + 1 )2 = 7 2. The following illustration could be used to reinforce why squaring individual terms of an equation is not the same as squaring both sides of the equation. 3+4=7 32 + 42 = 72 9 + 16 = 49 25 ≠ 49 134 3+4=7 (3 + 4)2 = 72 72 = 72 49 = 49 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Ask students to participate in Commit and Toss. Provide students with a selected response problem as shown below. They anonymously commit to an answer and provide a justification for the answer they selected. Students crumble their solutions into a ball and toss the papers into a basket. Once all papers are in the basket, ask students to reach in and take one out. Ask students to then move to the corner of the room designated to match their selected response. In their respective corners, they should discuss the similarities or differences in the explanations provided and report back to the class. 5.3 Radical Equations SB: pp.294-303 TR: pp.194-199 BLM: 5-3, 5-6 What is the value of x in the equation −3x + 6 = 6 ? (A) (B) (C) (D) -10 -2 0 10 Explain your reasoning: (AN3.1, AN3.2, AN3.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 135 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN3 Continued... Achievement Indicators: AN3.1, AN3.2 AN3.3, AN3.4 Continued The area model can also be used to solve radical equations. In Grade 8, students viewed the area of the square as the perfect square number, and the side length of the square as the square root (8N1). Recall that if a square has an area of 4, then its side has a length of 2. Similarly, if a square has an area of 3, then its side has a length of 3 . Teachers should prompt discussion about the side length of a square if its area is x. Consider the following example: Solve x − 7 = 3 . Using the area model, students label the dimensions of one square as x − 7 and the dimension of the other square as 3. They can then determine the area of each square. x–7 3 A = x–7 A = 9 = Students should recognize that the goal is to determine the value of x which results in the same area for both squares. Solving the equation x - 7 = 9 results in x = 16. This representation helps students visualize what each equation is describing. Encourage them to check their answers by substituting the value back into the original equation. Students will also solve radical equations that result in quadratic equations. Use an example such as x − 9 = 4 and the following questions to promote student discussion: 2 • What type of equation will this radical equation result in? • How is the radicand similar yet different to those studied to date? • How many solutions might this equation have? Using technology, present the graph of the corresponding function y = x − 9 for students to analyze. 2 y 10 8 6 y=4 4 2 -10 -8 -6 -4 -2 -2 2 4 6 8 10 x -4 From the graph, students should observe that the domain for the function is x ≥ 3 or x ≤ -3. This will help them determine whether x = ±5 are the solutions to the equation. 136 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Ask students to respond to the following: Pre-Calculus 11 The following steps show how a student solved the equation 3 + 2 n + 4 = 5 . Ask students if the final answer is correct and whether the student should receive full marks for the solution. They should justify their answer. 5.3 Radical Equations SB: pp.294-303 TR: pp.194-199 BLM: 5-3, 5-6 3+2 n +4 =5 5 n+4 =5 n +4 =1 ( n+4 ) 2 = 12 n +4 =1 n = −3 (AN3.1, AN3.2, AN3.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 137 RADICAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN3 Continued... Achievement Indicators: AN3.1, AN3.2 AN3.3, AN3.4 Continued Students should proceed to solve the radical equation algebraically by squaring both sides of the equation. They can solve the resulting quadratic equation using the method of their choice (i.e., graphing, factoring, completing the square or the quadratic formula). Students will also solve equations that involve two radical expressions. As students square both sides in an equation such as x + 7 = x + 1 they should recognize the resulting equation still contains a radical. Therefore, they will need to repeat the process of isolating the radical term and squaring both sides of the equation again. Continue to remind students of the importance of checking for extraneous roots when solving radical equations. AN3.5 Solve problems by modeling a situation using a radical equation. Students will be exposed to application problems where the equation may contain a radical that is a square root. They will solve for the unknown variable by squaring both sides of the equation. Provide students with the following example and ask them to answer the questions: • 138 Collision investigators can approximate the initial velocity, v, in kilometres per hour, of a car based on the length, l, in metres, of the skid mark. The formula v = 12.6 l + 8, where l ≥ 0 models the relationship. What length of skid is expected if a car is travelling 50 km/hr when the brakes are applied? How is knowledge of radical equations used to solve this problem? ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RADICAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • Ask students to solve each of the following equations. Remind them to check for extraneous roots. (i) (ii) 3x − 5 − 2 = 3 Pre-Calculus 11 5.3 Radical Equations SB: pp.294-303 TR: pp.194-199 BLM: 5-3, 5-6 2 x − 8x = 3 (iii) 2 x + 4 x = 3 − 2 x (AN3.1, AN3.2, AN3.3) • Ask students to answer the following: (i) The period T (in seconds) is the time it takes a pendulum to make one complete swing back and forth. This is modelled L by T = 2π 32 , where L is the length of the pendulum in feet. Ask students to determine the period of the pendulum if its length is 2 ft. (AN3.5) (ii) The radius of a cylinder can be found using the equation r = πVh where r is the radius, V is the volume, and h is the height. A cylindrical tank can hold 105.62 m3 of water. If the height of the tank is 2 m, what is the radius of its base? (AN3.5) (iii) The surface area (S) of a sphere with radius r can be found using the equation S = 4πr2. (a) Using the given equation, how could you find the radius of a sphere given its surface area? Write the equation. (b) The surface area of a ball is 426.2 cm2. What is its radius? (AN3.5) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 139 RADICAL EXPRESSIONS AND EQUATIONS 140 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Rational Expressions and Equations Suggested Time: 12 Hours RATIONAL EXPRESSIONS AND EQUATIONS Unit Overview Focus and Context In this unit, students will simplify a rational expression and determine the non-permissible values. They will perform the operations on rational expressions (addition, subtraction, multiplication, division). Students will solve problems that involve rational equations. They will determine the solution to a rational equation algebraically and identify the non-permissible values. Outcomes Framework GCO Develop algebraic reasoning and number sense. SCO AN4 Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). SCO AN5 Perform operations on rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). SCO AN6 Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials or trinomials). 142 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Mathematics 2200 Mathematics 3200 Algebra and Number Algebra and Number Relations and Functions AN5 Demonstrate an understanding of commonfactors and trinomial factoring,concretely, pictorially and symbolically. AN4 Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). RF14 Graph and analyze rational functions (limited to numerators and denominators that are monomials, binomials or trinomials). [C, ME, R] [CN, R, T, V] [C, CN, R, V] AN5 Perform operations on rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). [CN, ME, R] AN6 Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials or trinomials). [C, PS, R] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 143 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN4 Determine equivalent forms of rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials). In Grade 9, students solved problems that involved arithmetic operations on rational numbers (9N3). They will now be introduced to rational expressions limited to numerators and denominators that are monomials, binomials and trinomials. They will simplify them and determine the non-permissible values. [C, ME, R] Achievement Indicators: AN4.1 Explain why a given value is non-permissible for a given rational expression. AN4.2 Determine the nonpermissible values for a rational expression. A rational expression is any expression that can be witten as the quotient P( x ) of two polynomials, in the form Q ( x ) where Q(x) ≠ 0. To begin work with rational expressions, provide students with several examples of 2 x expressions, such as 45 , 2yx , xx +−14 , 5 , 2π , 2 y and ask them to identify and explain why an expression is or is not a rational expression. It should be pointed out to students that all rational expressions are algebraic fractions but not all algebraic fractions are rational expressions. In the above list, for example, 45 , 5 , and 2π are not rational expressions. Non-permissible values are the values of a variable that make the denominator of a rational expression equal zero. In Grade 7, students were introduced to the concept of why a number cannot be divided by zero (7N1). Students should first find the non-permissible values of a rational expression where the denominator is a first degree polynomial and then progress to second degree polynomials. Given the expression, x x + 2 , for example, use the following to promote student discussion around non-permissible values: • Using inspection, what value of x would make the denonimator zero? • Explain why this value of x is called a non-permisible value. • Fill in the following table of values. What do you notice? What is the domain of the expression x x+ 2 ? x y -3 -2 -1 0 1 3 • Write a general rule for determining the non-permissible value for a first degree denominator. • Can the numerator be equal to zero? Explain your reasoning. Students should notice that the non-permissible value of x can be written as x + 2 , x ≠ -2. 144 2 x x +2 is -2. This ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • Ask students to write a rational expression for the following non-permissible values of 0, -2 and 3. They should compare their answers with the class. Pre-Calculus 11 (AN4.2) Interview • Ask students to explain why x = 2 is a non-permissible value for 3x x −2 . (AN4.2) 6.1: Rational Expressions Student Book (SB): pp.310-321 Teacher Resource (TR): pp.209216 BlackLine Master (BLM): 6-4, 6-5 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 145 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN4 Continued ... Achievement Indicators: AN4.1, AN4.2 Continued A common error occurs when students generalize that the nonpermissible value is zero rather than looking at the value(s) of x that produce a denominator of zero. Encourage them to substitute the non-permissible value(s) for x back into the denominator to verify the denominator results in zero. Students should also be exposed to rational expressions where the denominator is a second degree polynomial. Discuss an expression, such as x2−1 . Ask students to answer the following questions: 3 x −12 • Using inspection, what value(s) of x would make the denominator zero? • What other strategies can be used to solve the quadratic equation? • Are there any rational expressions without non-permissible values? To solve 3x2-12, students may remove the greatest common factor (GCF) and apply the zero product property, apply the quadratic formula or the square root property. Question students as to which method is more efficient and why. If students use the square root property, they divide the equation 3x2 = 12 by 3 and solve x2 = 4. Some may mistakenly write the nonpermissible value as x = 2 rather than x = ±2. Another error occurs when students factor 3x2 - 12 = 0 as 3(x - 2)(x + 2) and include 3 as a nonpermissible value. They may also incorrectly factor the expression in the denominator, whether it is a binomial or trinomial. Remind them to verify their work by expanding the product of the factors using the distributive property. Students should be exposed to rational expressions with more than one variable. If the expression 2x - 3y is in the denominator, for example, the non-permissible values can be found by solving the equation 2x - 3y = 0 for x or y. Students have this option unless it is specifically stated in the problem to solve for a particular variable. 146 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Ask students to work in groups to participate in the Domino Game. Provide each group with10 domino cards. One side of the card should contain a rational expression, while the other side contains non-permissible values for a different rational expression. The task is for students to lay the dominos out such that the nonpermissible values on one card will match with the correct rational expression on another. They will eventually form a complete loop with the first card matching with the last card. A sample is shown below: 3x 2 x−1 x ≠ 13 x+2 3 x −1 x ≠ − 12 7x 2 x+1 6.1: Rational Expressions SB: pp.310-321 TR: pp.209-216 BLM: 6-4, 6-5 x≠2 (AN4.1, AN4.2) Journal • Ask students to respond to the following: x +3 What are the non-permissible values for x 2 − 16 ? I think the nonpermissible value is 4. I think the nonpermissible values are – 4 and 4. I think the nonpermissible values are – 4, -3 and 4. Who is correct? Justify your answer by solving the problem. (AN4.1, AN4.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 147 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN4 Continued ... Achievement Indicators: AN4.1, AN4.2 Continued It is important for students to differentiate between non-permissible values and inadmissible values. Remind students that inadmissible values were discussed in the Quadratics Unit when they used quadratic functions to model situations. These are values that do not make sense in a given context. Students will continue to work with inadmissible values for a variable in a rational expression. If a boat traveled 20 km with a speed of x km/h, for example, the time taken for the trip would be represented by . If students 20 are asked to determine the slowest x speed the boat can travel, they should recognize that the non-permissible value is 0 but the inadmissible value is x < 0. AN4.3 Compare the strategies for writing equivalent forms of rational expressions to the strategies for writing equivalent forms of rational numbers. In Grade 7, students developed skills in writing equivalent positive rational numbers (7N7). Students will apply these strategies to rational expressions. This concept is essential when adding and subtracting rational expressions later in this unit. AN4.4 Determine a rational expression that is equivalent to a given rational expression by multiplying the numerator and denominator by the same factor (limited to a monomial or a binomial), and state the non-permissible values of the equivalent rational expression. Students should recognize that they can multiply or divide a rational expression by 1 without changing its value. A rational expression is not equivalent to another rational expression if their restrictions are different. Guide students through the following activity: • Consider the rational number 23 . Ask students to write a rational number by multiplying both the numerator and denominator by 3, by 5, and by -4? Did the value of their fraction change? • Consider the rational expression x4 where x ≠ 0. Ask students to write a rational expression by multiplying both the numerator and denominator by 2, by x, and by x + 1. Did any of their expressions produce a new restriction? Using substitution, ask students to verify if the expressions are equivalent. When the expressions x4 , x ≠ 0 and 4( x +1) , x ≠ 0,-1 are x ( x +1) compared, they are both undefined at x = 0. When x = -1, however, the expression x4 simplifies to -4 while the expression x4(( xx ++1) is undefined. 1) Since the expressions are not equal for the same value of x, the expressions are not equivalent. Although graphing rational functions is not an outcome in this course, teachers could, as an alternative, prove rational expressions are equivalent for all permissible values of the variable by showing students the graphs of these functions. 148 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to complete the following table: Are the expressions equivalent? x +3 x −4 5 x −5 x +2 x −3 and and and Yes No Justify your choice SB: pp.310-321 TR: pp.209-216 BLM: 6-4, 6-5 4 x + 12 4 x − 16 5x + 25 2 x − 25 3x + 6 2x − 6 (AN4.3, AN4.4) Journal • 6.1: Rational Expressions x +1) Your friend thinks the expressions x2−x3 and ( x2−x3)( are ( x +1) equivalent. Ask students to explain why these expressions are not equivalent? (AN4.3, AN4.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 The resource Pre-Calculus 11 discusses equivalent rational expressions on p.313. It is important to note that 7s 7s 2 s − 2 and s ( s − 2 ) are not equivalent rational expressions since a new restriction has been added. 149 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN4 Continued ... Achievement Indicators: AN4.3, AN4.4 Continued When writing an equivalent expression, caution students to use the x distributive property appropriately. When simplifying x + 4 × 22 , for example, students may incorrectly write 2 x2 x+ 4 or x2+x8. To avoid this error, encourage students to place brackets around the binomial when multiplying. AN4.5 Simplify a rational expression. Simplifying a rational expression to lowest terms mirrors the process of simplifying fractions. In both cases, common factors in the numerator and denominator form a ratio of one and can be simplified. AN4.6 Explain why the nonpermissible values of a given rational expression and its simplified form are the same. Discuss with students the benefit of simplying rational expressions, whether it be for evaluating or performing operations. Ask students 2 to evaluate the expression x +x 4 x where x ≠ 0 at x = 2. Then evaluate the expression x + 4 where x ≠ 0 at x = 2. Students should answer the following questions: ( ) • What is the result when substituting the value into the original unsimplified expression? • What is the result when substituting the value into the simplified expression? • Why were the results the same? • What is the benefit of simplifying an expression before substituting values for the variables? • Why does the simplified expression include a non-permissible value? Students should recognize one of the benefits of simplifying an expression is to create an equivalent expression that is easier to evaluate. Ask them why the domain of a rational expression is always determined before and after the expression is simplified. A simplified rational expression, for example, may not have any non-permissible values. However, the simplified expression must retain the non-permissible values of the original expression for both to be equivalent. 150 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Ask students to participate in the following activities: (i) Divide the class into two groups. One group will be given rational expressions and the other group will be given the associated rational expression in simplest form. Ask students to find a partner who has a rational expression equivalent to theirs. (AN4.5) (ii) Students create a unique Bingo card for Rational Expression Bingo. Distribute a blank Bingo card to each student. Teachers should predetermine various expressions involving rational expressions they would like students to simplify. The expressions should be placed in a bag, with simplified expressions on the board. Ask students to write one of the simplified expressions in each sqaure. The center square should remain a "free" space. The teacher pulls an expression from a bag. Students simplify the expression, find its value on their card and cross it off. The first person with a straight line or four corners wins, or the first person with an X or a T on the Bingo card could win. (AN4.5) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 6.1: Rational Expressions SB: pp.310-321 TR: pp.209-216 BLM: 6-4, 6-5 151 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN4 Continued... Achievement Indicator: AN4.7 Identify and correct errors in a given simplification of a rational expression, and explain the reasoning. It is beneficial to have students analyze solutions that contain errors. Along with providing the correct solutions, they should be able to identify incorrect solutions, including why errors might have occurred and how they can be corrected. When simplifying rational expressions, students may cancel terms x 2 +x x 2 +x rather than factors. They may simplify, for example, x 2 −1 as x 2 −1 resulting in -x. To help students see this error ask them to make a 8 = 55++73 and comparison with a numerical rational expression such as 12 8 = 55++73 = 73 . Ask them if 128 is equal to 73 . Students should realize 12 that cancelling a portion of the factor is incorrect. Another error occurs when students omit a numerator of 1 after the rational expression is simplified. They mistakenly simplify 6x3 , for example, as 2x. Encourage students to check the reasonableness of their answer by rewriting the the expression as 63 i x1 . 152 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Observation Authorized Resource • Pre-Calculus 11 Set up centres containing examples of incorrect simplified rational expressions and their non-permissible values. Ask students to move around the centres to identify and correct the errors. A sample is shown below: 8 x − 12 , x ≠ 0, 23 2 6x − 4 x 4(2 x − 3) 6.1: Rational Expressions SB: pp.310-321 TR: pp.209-216 BLM: 6-4, 6-5 2 x (3x − 2) 4 (1) 2x 2 x , x ≠ 0, 23 (AN4.7) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 153 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN5 Perform operations on rational expressions (limited to numerators and denominators that are monomials, binomials or trinomials. In Grade 9, students solved problems involving operations on rational numbers (9N3). This will now be extended to adding, subtracting, multiplying and dividing rational expressions with numerators and denominators limited to monomials, binomials and trinomials. [CN, ME, R] Achievement Indicators: AN5.1 Compare the strategies for performing a given operation on rational expressions to the strategies for performing the same operation on rational numbers. AN5.2 Determine the nonpermissible values when performing operations on rational expressions. Multiplying and dividing rational expressions is very similar to the process students used to multiply and divide rational numbers. Using 2 examples such as 12 × 10 and xx2 −−49x × xx −−43 , ask students to simplify 25 21 and find the product for each. They should think about whether the strategy for multiplying rational expressions is the same as the strategy for multiplying rational numbers. Ask them to also consider at what step the non-permissible values are determined. It is important for students to recognize the importance of factoring the numerator and denominator of the rational expression, if possible, before the product is determined. Ask students to answer the following: • AN5.3 Determine, in simplified form, the product or quotient of rational expressions. 2 Find the product of x x2 −4 × x x+ 2 using two different strategies. Which strategy is more efficient? Why? Reinforce that multiplication of rational expressions follows the same procedure as multiplying rational numbers, but with the added necessity of determining the non-permissible values for the variables. Provide an opportunity for students to compare the division of rational numbers to division of rational expressions. Students sometimes forget to identify the non-permissible values for the numerator of the divisor in a division statement. Reinforce the importance of this step through the use of examples. 154 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to create an activity sheet where the column on the left contains operations with rational expressions and the column on the right contains the non-permissible values (not in the same order). Students will then exchange their sheets. The task is to match each expression with its correct non-permissible values. (the non-permissible values may match more than one expression on the left and may not match any). (AN5.2) • 6.2: Multiplying and Dividing Ask students to work in groups to complete the following table. Rational Expressions Students should explain the similarities between finding the lowest common denominator (LCM) of two rational numbers versus two SB: pp.322-330 TR: pp.217-223 rational expressions. BLM: 6-4, 6-6 Rational LCM Rational Expres- LCM Similarities Number sion 4 5 1 5 + − 7 12 3 6 5 2x − 1 7 4x 15 x −3 + 3 8 − + 2x − 1 5 6x − 18 2 2 −2 x − 36 + 4 3x + 18 (AN5.1) Journal • Sean stated that the permissible values for the quotient and the 2 2 8 product of the expressions x x+2 4−x16+ 3 and xx2 −−72xx+−12 are the same. Ask students if they agree or disagree with his statement. They should justify their answer. (AN5.2, AN5.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 155 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN5 Continued ... Achievement Indicators: AN5.4 Determine, in simplified form, the sum or difference of rational expressions with the same denominator. AN5.5 Determine, in simplified form, the sum or difference of rational expressions in which the denominators are not the same and which may or may not contain common factors. AN5.2 Continued In Mathematics 1201, students were introduced to the lowest common multiple for a set of numbers (AN1.3). Students should compare finding the lowest common denominator of rational numbers to finding the lowest common denominator of rational expressions. Allow students to discover the different situations that occur when finding the lowest common denominator of two numbers and then compare this to rational expressions. Consider the following table: Rational Number Situation Rational Expression the denominators are the 3 2 x2 − − 1 7 7 x +1 x +1 same one denominator is a 1 3 +5 − 1 12 6 x +5 4 x + 20 multiple of the other the denominators have no 3 + 4 2 7 + 2x x −1 3 2 common factors the denominators have a 7 5 + 1 +1 x 2 −9 4 x +12 14 6 common factor Ask students to answer the following questions related to the rational expressions: • How do you find the lowest common denominator? Why is it beneficial to simplify the expression before finding the lowest common denominator? • What are the non-permissible values? • Can you list other examples that fit each situation? Similar to rational numbers, rational expressions can be added if they have common denominators. Once students determine the lowest common denominator, they should rewrite each rational expression with that common denominator. A common student error involves adding or subtracting the numerators without first writing the fractions with a common denominator. For example, students mistakenly add x5 + 23 as x 8+ 2 . Remind students to be careful when subtracting rational expressions. They sometimes forget to distribute the negative sign when there is more than one term in 3x −2 2 x −4 the numerator. For example, ( x + 2 )( x − 2 ) − ( x + 2 )( x − 2 ) is often written as 3 x − 2 − 2 x −4 ( x + 2 )( x − 2 ) . Encourage students to use brackets to help them avoid this mistake. 156 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to simplify the rational expressions: (i) x +7 2 x +14 (ii) 2 x −6 x 2 − x −6 − −35xx− 21 − 6.3: Adding and Subtracting Rational Expressions 3 x +12 x 2 + x −12 (AN5.5) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SB: pp.331-340 TR: pp.224-231 BLM: 6-4, 6-7 157 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN5 Continued ... Achievement Indicator: AN5.6 Simplify an expression that involves two or more operations on rational expressions. A complex fraction is an example of an expression involving two or more operations on a rational expression. In order to avoid errors, students should place brackets appropriately and use the order of operations correctly. Provide examples to students illustrating various strategies to simplify an expression containing a complex fraction. Students may first simplify both the numerator and denominator, invert and multiply and then simplify. Consider the example : 1 + 1 x+2 x −2 x + x +1 2 x2 −4 = x −2 ( x + 2)( x − 2) x ( x + 2)( x − 2) x+2 + ( x + 2)( x − 2) + x −2 ( x + 2)( x − 2) x − 2) 2x = ( x + 2)( × ( x +22)( = x − 2) x−2 2x 2 x −2 = x x −1 As an alternative, students can multiply the entire expression by the common denominator divided by itself. This common denominator is obtained by considering all existing denominators in the expression. ⎛ x 1+ 2 + x 1−2 ⎞ ( x + 2)( x −2) = ⎜ x ⎟ ⎜ 2 + 1 ⎟ ( x + 2)( x −2) ⎝ x −4 x + 2 ⎠ = 1 1 x + 2 ( x + 2)( x − 2) + x − 2 ( x + 2)( x − 2) x 1 ( x − 2)( x + 2) ( x + 2)( x − 2) + x + 2 ( x + 2)( x − 2) x − 2 +x + 2 x + x −2 = 2x 2 x −2 = x x −1 As students work through the two possible strategies, they should think about efficiency and what will work best for any complex fraction. 158 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Presentation Authorized Resource • Pre-Calculus 11 Ask students, working in pairs or small groups, to create two rational expressions. The first rational expression should contain each of the operations. The other expression would involve two or more operations on rational expressions. They should solve the expressions and present their findings to the class. (AN5.3, AN5.4, AN5.5, AN5.6) 6.3: Adding and Subtracting Rational Expressions SB: pp.331-340 TR: pp.224-231 BLM: 6-4, 6-7 Paper and Pencil • Ask students to find the area of the shaded region. (AN5.6) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 159 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN6 Solve problems that involve rational equations (limited to numerators and denominators that are monomials, binomials or trinomials). [C, PS, R] In Grade 9, students solved linear equations (9PR3). Previously in this course, students solved quadratic equations and identified inadmissible roots (RF5). They will now solve equations containing rational expressions and check if the solutions are permissible values. Roots that are non-permissible are extraneous. (RF2). It is intended that the rational equations be those that can be simplified to linear and quadratic equations. Achievement Indicators: AN6.1 Determine the nonpermissible values for the variable in a rational equation. AN6.2 Determine the solution to a rational equation algebraically, and explain the strategy used to solve the equation. AN6.3 Explain why a value obtained in solving a rational equation may not be a solution of the equation. Students will be exposed to different strategies when solving rational equations. It would be beneficial to begin with an example that is easier to visualize before moving on to more complex equations. Some x = 52 . students may use trial and error to solve an equation such as 10 Others may be able to determine the solution by inspection. Encourage students to discuss their ideas. For example, a student may respond that in order to get the number 10, the number 5 must be doubled therefore 2 is also doubled resulting in x = 4. This student response is a great lead into the strategy of creating an equivalent rational equation with common denominators. Ask students to rewrite the rational equation x 4 = 10 with a common denominator ( 10 ) and then write an equation with the numerators. Another strategy involves eliminating the denominators. Use an x = 52 , to promote discussion around lowest common example such as, 10 denominator: • What is the lowest common denominator of 10 and 5? • What would happen if the lowest common denonimator was multiplied on both sides of the equation? Why is this mathematically correct? • What is the simplified equation? • What is the solution? Once the strategies have been discussed, students will then be exposed to solving more complex rational equations. 160 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Provide each student with a rational equation. Ask them to identify the non-permissible roots and then solve the equation. Ask students not to write their name on the paper since student solutions will be collected and then redistributed randomly around the room. Ask students to verify if the solution is correct. If the solution is incorrect, they will identify the error and write the correct solution. (AN6.1, AN6.2) 6.4: Rational Equations SB: pp.341-351 TR: pp.232-238 BLM: 6-4, 6-8 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 161 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN6 Continued ... Achievement Indicators: AN6.1, AN6.2 AN6.3 Continued When solving x3 + 27x = 51 , for example, students may find a common denominator for the left side of the equation and then proceed to solve 13 = 51 . They can also multiply both sides of the equation by the 2x lowest common denominator (10 x ) x3 + (10 x ) 27x = (10 x ) 51 . It is important, however, for students to recognize that they can reduce the numbers of steps by multiplying both sides of the original equation by the lowest common denominator. Students can add or subtract the terms on the left hand side or the right hand side of the equation before they cross multiply. This process, however, may lead to an equation where the degree of the polynomial is greater than what they started with. Consider the following example: 2 x 2 +1 x +3 = x 4 2 x 2 +1 x +3 = x 2 + 3 x + 20 4( x + 3) 2 + 5 x +3 2 4( 2 x + 1)( x + 3) = ( x + 3x + 20)( x + 3) This example results in a cubic equation. Students are only familar with solving quadratic equations at this point. Therefore, multiplying both sides of the equation by the lowest common denominator would be the method students would choose. Caution students that it is necessary to find the non-permissible roots at the beginning of the solution since some rational equations may lead to extraneous roots. Consider the equation 2xx++53 + 21 = 2(−x14+5) . Ask students to answer the following questions: • What is the non-permissible root? What does this mean? • What is the solution to the resulting linear equation? • Why is it important to check the solution by using the original equation? Students should recognize solutions that are non-permissible values are extraneous roots. Therefore, they must be eliminated as a valid solution. When solving rational equations, the modified equation may result in either a linear or quadratic equation. Students will have a choice whether to use the quadratic formula or their factoring skills to solve the quadratic equation. Remind them to verify their solutions to avoid extraneous roots. 162 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Journal Authorized Resource • Pre-Calculus 11 Ask students to reflect on the process of solving a rational equation. They should respond in writing to three reflective prompts providing six responses, as shown below, to describe what they learned. 3 things I understand: 1. 6.4: Rational Equations SB: pp.341-351 TR: pp.232-238 BLM: 6-4, 6-8 2. 3. 2 things I am still struggling with: 1. 2. 1 thing that I will work on: 1. (AN6.1, AN6.2, AN6.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 163 RATIONAL EXPRESSIONS AND EQUATIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN6 Continued ... Achievement Indicator: AN6.4 Solve problems by modeling a situation using a rational equation. Students will be expected to write an equation to represent a problem. They should be exposed to examples such as the following: 5 The sum of a number and its reciprocal is 2 . Students can begin this example using trial and error and discuss possible solutions. They should then proceed to write the rational equation x + x1 = 52 . Encourage students to write their own example and share with the class. It is important for students to recognize that inadmissible roots come from the context of the problem. Discuss different scenerios that produce inadmissible roots. A negative numerical value, for example, would not make sense if referring to time, height and length. Students may have difficulty interpreting the information from the word problem and writing the rational equation. Encourage them to use tables and diagrams to help them break down the information. Consider the following example: Sherry mows a lawn in 4 hours. Mary mows the same lawn in 5 hours. How long would it take both of them working together to mow the lawn? Pose the following questions to begin a discussion: • How much of the lawn would Sherry mow in 1 hour? • How much of the lawn would Mary mow in 1 hour? • How much of the lawn would both mow together in 1 hour? Completing a table such as the one below should help students organize their information. Sherry Time to mow lawn (hours) 4 Fraction of lawn mowed in 1 hr Mary 5 1 5 Both x 1 x 1 4 When solving the equation 41 + 51 = x1 , encourage students to check that the solutions satisfy the original equation are permissible, and in the case of a word problem, realistic in the context. Students should also be exposed to word problems that produce a rational equation resulting in solving a quadratic equation. In such cases, there may be an inadmissible value that will need to be rejected in the context of the problem. 164 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 RATIONAL EXPRESSIONS AND EQUATIONS General Outcome: Develop algebraic reasoning and number sense. Suggested Assessment Strategies Resources/Notes Performance Authorized Resource • Pre-Calculus 11 Create pairs of cards with word problems and matching equations to solve the word problems. Distribute the cards amongst the students and have them find their partner by matching the word problem with the corresponding equation. Once they have found their partner, students should work in pairs to solve the equation and verify their solution. 6.4: Rational Equations SB: pp.341-351 TR: pp.232-238 BLM: 6-4, 6-8 (AN6.2, AN6.4) Paper and Pencil • A student was given the following word problem: It takes Mike 9 hours longer to construct a fence than it takes Jason. If they work together, they can construct the fence in 20 hours. How long would it take Mike to construct the fence alone? 20 The student solved the equation 20 t + t +9 = 1 and stated the solutions to the word problem were 36 and -5. Ask students to verify the solution and state whether the student is correct. (AN6.2, AN6.3, AN6.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 165 RATIONAL EXPRESSIONS AND EQUATIONS 166 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Absolute Value and Reciprocal Functions Suggested Time: 16 Hours ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Unit Overview Focus and Context In this unit, students will focus on absolute values, determine the absolute value of numerical expressions, and solve problems involving the absolute value. They will graph the absolute value of a linear function and the absolute value of a quadratic function. Students will identify characteristics of the graph, including the intercepts and the domain and range. They will also represent the absolute value function using piecewise notation. Students will solve absolute value equations graphically and algebraically. Students will expore reciprocal functions by comparing the graphs of a function and its reciprocal. The focus will be on linear and quadratic functions. Outcomes Framework GCO GCO Develop algebraic reasoning and number sense. Develop algebraic and graphical reasoning through the study of relations. SCO AN1 Demonstrate an understanding of the absolute value of real numbers. SCO RF2 Graph and analyze absolute value functions (limited to linear and quadratic functions) to solve problems. SCO RF11 Graph and analyze reciprocal functions (limited to the reciprocal of linear and quadratic functions). 168 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Mathematics 2200 Algebra and Number AN1 Demonstrate an understanding of the absolute value of real numbers. [R, V] Relations and Functions Relations and Functions RF5 Determine hte characteristics RF2 Graph and analyze absolute of the graphs of linear relations, value functions (limited to linear including the: and quadratic functions ) to solve problems. • intercepts [C, PS, R, T, V] • slope Mathematics 3200 Relations and Functions RF14 Graph and analyze rational functions (limited to numerators and denominators that are monomials, binomials or trinomials). [CN, R, T, V] • domain • range [CN, PS, R, V] RF6 Relate linear equations expressed in: RF11 Graph and analyze reciprocal functions (limited to the reciprocal of linear and quadratic functions). [CN, R, T, V] • slope-intercept form (y = mx + b) • general form (Ax +By+C=0) • slope-point form (y-y1=m(x-x1)) to their graphs [CN, R, T, V] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 169 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Algebra and Number Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to AN1 Demonstrate an understanding of the absolute value of real numbers. [R, V] Achievement Indicators: AN1.1 Determine the distance of two real numbers of the form ±a, a ∈ \ , from 0 on a number line, and relate this to the absolute value of a ( a ). AN1.2 Determine the absolute value of a positive or negative real number. AN1.3 Explain, using examples, how distance between two points on a number line can be expressed in terms of absolute value. AN1.4 Determine the absolute value of a numerical expression. AN1.5 Compare and order the absolute values of real numbers in a given set. 170 The concept of absolute value is new to students. They will determine the absolute value of numerical expressions and compare and order the absolute values of real numbers in a given set. Introduce students to the absolute value of a number as its distance from zero. Ask them to plot the integers 5 and -5 on a number line. They should notice, when graphed, that these numbers are are the same distance from zero. It is important for students to understand that absolute value only asks “how far?”, not “in which direction?”. Reinforce that distance is always positive. This leads to the definition of the absolute value of any real number a: ⎧a , if a ≥ 0 a =⎨ ⎩−a , if a < 0 Students will extend the concept of absolute value to include the distance between any two real numbers. This could be first investigated using natural numbers and then extended to integers and real numbers. Completing a table such a the one below should help students recognize that the distance between a and b can be represented by |a - b| or |b - a|. a b 2 -5 2.68 6 -10 5.75 distance between a and b 4 value of |a - b| |2-6|=|-4|=4 value of |b - a| |6-2|=|4|=4 In Grade 9, students simplified numerical expressions using the order of operations (9N4). This is now extended to include expressions containing absolute value. Ask students to compare the expressions 3 - 4(2) and |3 - 4(2)|. They may have difficulty determining where the absolute value of a numerical expression occurs in the order of operations. They will simplify the expression inside the absolute value symbol using the order of operations and then take the absolute value of the resulting expression. Students will compare and order absolute values of real numbers. If the absolute value contains a fraction, students may find it helpful to change it to a decimal representation. Once students evaluate and compare the absolute values in a given set, they can place the values on a number line to help them order the set. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic reasoning and number sense Suggested Assessment Strategies Resources/Notes Journal Authorized Resource • Pre-Calculus 11 Provide students a fictitious town with one main street. This street should contain at least eight landmarks located to the right and left of the town square which is representative of the origin. Each landmark should be given a name (gas station, library, etc.) Ask students to pose three questions where the distance between any two landmarks is requested.They should then answer the questions. (AN1.3) 7.1 Absolute Value Student Book (SB): pp.358-367 Teacher Resource (TR): pp.250255 Blackline Master (BLM): 7-3, 7-4 Observation • Create a set of cards with a variety of examples involving the absolute values of real numbers and numerical expressions. In groups of three, each student will receive five cards. They will place the cards in postition that produces an ordered set of cards. When each member of the group is finished, ask each student to explain their resoning to the other members. (AN1.2, AN1.4, AN1.5) • Ask students to create a human number line. Each student is given a card containing an absolute value of a real number or numerical expression. They order themselves into a line based on the relative size of their number. Ask them to expain why they chose their position. As an alternative, a skipping rope or piece of string could be used. Students attach their number to the line in the appropriate position. (AN1.2, AN1.4, AN1.5) Performance • Divide the class into teams consisting of about five students. Each team would line up one behind the other. Provide each team with a list of absloute value problems to simplify. The first student in each line would write the answer to one of the absolute value problems using mental math skills. They would then move to the back of the line and the second student would repeat this with a different problem. The first team to complete their questions with the correct answers would be the winners. (AN1.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Web Link The following website is an interactive game for ordering the absolute values of real numbers. http://www.sheppardsoftware.com (search: number balls absolute value) 171 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF2 Graph and analyze absolute value functions (limited to linear and quadratic functions) to solve problems. [C, PS, R, T, V] In Grade 9, students solved linear equations (9PR3). In Mathematics 1201, students analyzed the graphs of linear relations (RF6). Earlier in this course, students analyzed quadratic functions in vertex and standard form to identify the characteristics of the corresponding graph (RF3, RF4). They will now graph and analyze absolute value functions, limited to linear and quadratic functions. Achievement Indicators: RF2.1 Create a table of values for y = |f (x)|, given a table of values for y = f (x). RF2.2 Sketch the graph of y = |f (x)|; state the intercepts, domain and range; and explain the strategy used. Students will compare the graph of a linear function to its corresponding absolute value function, using a table of values. They should identify the similarities and differences between y = |ax + b| and y = ax + b. Ask students to graph y = |x + 3|, for example, using the table of values for y = x + 3. To get an accurate picture of the absolute value graph, it is important to include appropriate x-values that will produce negative y-values. y y 6 6 5 5 4 4 3 3 2 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 1 2 3 4 5 6 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 1 2 3 4 5 6 x Using the graph as a visual aid, students should recognize that the section of the graph which lies above the x-axis remains the same and the section which lies below the x-axis is reflected in the x-axis. Ask students to identify characteristics of the absolute value graph, such as the intercepts, domain and range. As they look for the similarities and differences between the graphs, they should consider the following: • Is the x-intercept significant? • Is the domain dependent on the x-intercept? Explain. • Why is the domain of the the function the same as the domain of the absolute value function? Why is the range different? Comparing the graphs of y = |ax + b| and y = ax + b, students recognize the resulting shape of the graph. Using this shape, along with the x-and y-intercepts, they should be able to draw the graph. 172 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Performance • Ask students to work in groups for this activity. Each group will be given a different tables of values. They will create a table of values for y = |f (x)|. Ask students to display their tables around the classroom. They should observe and analyze the data. Each group will then present their conclusions to the rest of the class. Pre-Calculus 11 7.2 Absolute Value Functions SB: pp.368-379 TR: pp.256-261 BLM: 7-3, 7-5 (RF2.1) • Working in small groups, students will play the Absolute Matching Game. Each group will be given several cards. Half of the cards will have the graph of y = f (x) while the other half will have the graph of y = |f (x)|. The object of the game is to be the first group to pair up each graph with its absolute value graph. Note: For this game to be most effective, the graphs should have similar characteristics, such as the same x-intercepts but different slopes/vertices. (RF2.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 173 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF2 Continued ... Achievement Indicators: RF2.1, RF2.2 Continued Students will then proceed to work with and analyse the absolute value of quadratic functions. They will compare the graphs of the absolute value of a quadratic function to its original graph using a table of values. Ask students to graph, for example, y = |(x - 2)2 - 4| using the table of values for y = (x - 2)2 - 4. Remind them that the graph of a quadratic function can be obtained using vertex and the x-intercepts, as previously introduced in the Quadratic Functions unit. y y 6 6 5 5 4 4 3 3 2 2 1 1 -5 -4 -3 -2 -1 1 -1 2 3 4 x 5 -5 -4 -3 -2 -1 1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 2 3 4 5 x Students should notice that the graph of y = |f (x)| reflects the negative part of the graph of y = f (x) across the x-axis, while the positive part remains unchanged. Students should again note the similarities and differences between the two graphs. RF2.3 Generalize a rule for writing absolute value functions in piecewise notation. Students will develop a rule for writing absolute value functions in piecewise notation using a graph. Explain to students that piecewise functions are used to describe functions that contain distinct functions over different intervals. The graph of a linear absolute value function, for example, consists of two separate linear functions. The domain for each interval is dependent on the x-intercept(s). Revisit the graphs of y = x + 3 and y = |x + 3|. y y 6 6 5 5 4 4 3 3 2 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 1 1 2 3 4 5 6 x -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 -1 -2 -2 -3 -3 -4 -4 -5 -5 -6 -6 1 2 3 4 5 6 x Students should notice the turning point on the absolute value graph is located at the x-intercept. For values of x where y is positive, the graphs of y = |x + 3| and y = x + 3 are the same. For values of x where y is negative, the graph of y = |x + 3| is a reflection of the graph of y = x + 3. 174 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • As a pre-assessment, ask students to complete the following flowchart to describe what they know about linear and quadratic functions. Linear Example y = 3x − 4 Characteristics Pre-Calculus 11 7.2 Absolute Value Functions SB: pp.368-379 TR: pp.256-261 BLM: 7-3, 7-5 Graph Quadratics Example 2 Characteristics Graph y = ( x − 3) − 4 (RF2.1, RF2.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 175 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF2 Continued ... Achievement Indicators: RF2.3 Continued The absolute value function y = |x + 3| can be written in piecewise notation as: ⎧x + 3, if x ≥ −3 y =⎨ ⎩−( x + 3), if x < −3 This can be generalized to all absolute value functions: ⎧ f ( x ), where f ( x ) ≥ 0 y=⎨ ⎩− f ( x ), where f ( x ) < 0 Similarly, students can use the visual representation of the absolute value of a quadratic function to determine the piecewise function. This would be a good opportunity to also expose students to an algebraic method using sign diagrams to analyse where the quadratic function is positive or negative. Ask students to write, for example, the function y = |(x - 2)2 - 4| using piecewise notation. They will first use the quadratic function y = (x - 2)2 - 4 to determine the x-intercepts. They will then determine the distinct intervals where the function is positive and where it is negative on a number line. y 6 5 4 3 2 1 -3 -2 -1 1 -1 2 3 4 5 x -2 -3 -4 -5 -6 y-values + -3 -2 – y-values -1 0 1 2 y-values + 3 4 5 Using the sign diagram, students can write the piecewise function as: 2 ⎪⎧( x − 2) − 4, x ≤ 0 or x ≤ 0 y=⎨ 2 ⎪⎩−[( x − 2) − 4], 0 < x < 4 This can also be done without the graph as a visual aid. Relating the sign diagram to the x-axis of the graph, students could substitute an x-value from each interval into the function to determine where the function is positive or negative. When graphing the absolute value of a linear or quadratic function, they should recognize the shape of the graph, the y-intercept and the x-intercept(s). Using the information from the sign diagram, they can then draw the graph of the absolute value function. 176 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Performance • In small groups, ask students to participate in a card game, Piecewise Pairs, involving graphs and piecewise functions. Each group will be given several cards. Half the cards contain a graph while the other half will have a piecewise function. The object of the game is to be the first group to match each graph with its correct piecewise notation. Pre-Calculus 11 7.2 Absolute Value Functions SB: pp.368-379 TR: pp.256-261 BLM: 7-3, 7-5 Note: Graphs should be similar enough to each other so that students cannot just match them without any thought (e.g., have several graphs with the same x-intercept so additional analysis is involved). (RF2.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 177 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF2 Continued... Achievement Indicator: RF2.4 Solve an absolute value equation graphically, with or without technology. Students will explore the solutions to absolute value equations, focusing first on the graphical representation and then moving to an algebraic solution. Remind them of the properties of absolute value: • For a real number a, the absolute value |a| is the distance from a to the origin. • For two real numbers a and b, |a - b| is the distance between a and b on the number line. Students should first work with the absolute value of a linear equation before moving on to the absolute value of a quadratic equation. Ask students what it means to solve an equation such as |x - 2| = 6. They should be looking for points whose distance from 2 is 6. Using a number line, they should realize that both -4 and 8 are at a distance of 6 from 2. This reasoning will allow students to better understand the solutions when using a graph. Ask students to graph the function y = |x - 2| and y = 6 on the same coordinate plane to determine the points of intersection. y 10 y = |x – 2| 8 y=6 6 4 2 -10 -8 -6 -4 -2 2 4 6 8 10 x -2 They should notice the x-coordinates of the points of intersection are the solutions to the equation. Encourage students to verify the solution by substituting the values back into the equation. Similarly, this graphical approach can be used to solve the absolute value of quadratic equations. 178 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Pre-Calculus 11 7.2 Absolute Value Functions SB: pp.368-379 TR: pp.256-261 BLM: 7-3, 7-5 7.3 Absolute Value Equations SB: pp.380-391 TR: pp.262-267 BLM: 7-3, 7-6 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 179 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF2 Continued... Achievement Indicator: RF2.5 Solve, algebraically, an equation with a single absolute value, and verify the solution. Students will move from the graphical representation of a solution to an algebraic method to solve absolute value equations. It is important for students to begin with simpler absolute value equations and then move on to more complex equations. Remind students that the definition of an absolute value consists of two parts: where f (x) ≥ 0 and f (x) < 0. Consider the previous example |x - 2| = 6. The equation is split into two possible cases: ± (x - 2) = 6. Students will solve the equations (x - 2) = 6 and -(x - 2) = 6 to result in x = -4 and x = 8. Each solution should be checked for extraneous roots. When students solve absolute value equations involving a quadratic expression, such as |x2 - 4| = 3x, they may have to solve the resulting quadratic equation using their factoring skills or the quadratic formula. Encourage them to verify the solutions by substituting the x-values back into the equation and making sure the left hand side of the equation is equal to the right hand side of the equation. RF2.6 Explain why the absolute value equation |f(x)|< 0 has no solution. RF2.7 Determine and correct errors in a solution to an absolute value equation. RF2.8 Solve a problem that involves an absolute value function. 180 Remind students the absolute value of a number is always greater than or equal to zero. Students can use inspection to see if this property applies. Consider the example |x - 2| = -6. Ask students if their is any possible x value that can be substituted into the equation to result in a negative value. Students should be provided with worked solutions containing errors to a number of absolute value equations and asked to identify and correct the errors. Common errors include: • Treating the absolute value sign like parentheses. • Multiplying a constant by the expression within the absolute value sign (e.g., -2|x - 3| = |-2x + 6|). • Incorrectly placing the negative in front of the variable rather than the entire expression (i.e., When solving |x - 3| = 8, students may wirte -x - 3 = 8 instead of -(x - 3) = 8). • Not identifying extraneous roots. • Errors when using the quadratic formula. Problem solving using linear absolute value functions should be embedded throughout the unit and situated in a variety of contexts. In some problems, students will be given an absolute value function and asked to analyze it. In other problems, students will be required to create the function from the given information. Encourage students to check their answers and identify extraneous solutions. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 7.3 Absolute Value Equations Ask students to illustrate with a graph why |x - 2| = -6 has no solution. (RF2.6) • Place cards on the classroom wall consisting of linear and quadratic absolute value equations. Ask students to choose two cards, one linear and one quadratic. They should solve the equations and pass them in as exit cards. These exit cards can then be used later for students to determine and correct any errors. (RF2.5, RF2.7) • For the activity Pass the Problem, each group of students is given absolute value equations to solve algebraically. After a specific amount of time, ask students to swap their problem with another pair. If the group finished the problem, the other group will check the solution. If errors are identified the group will correct the error and then continue to complete the problem. If the group, however, did not finish answering the problem, the other pair will check the partially completed solution and pick up from where the group left off. When they are finished, they should share the completed responses with each other, defending their reasons for any changes they made and provide feedback on each other's thinking. SB: pp.380-391 TR: pp.262-267 BLM: 7-3, 7-6 Web Link The following sites provides interactive games related to absolute value. http://www.crctlessons.com/absolutevalue-equations-game.html http://www.crctlessons.com/absolutevalue-equations-game.html (RF2.5, RF2.7) Presentation • Ask groups of students to research a problem that incorporates the absolute value function. They should present the problem to the class, including the solution. (RF2.8) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 181 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF11 Graph and analyze reciprocal functions (limited to the reciprocal of linear and quadratic functions). [CN, R, T, V] Achievement Indicators: RF11.1 Compare the graph of y = f (1x ) to the graph of y = f (x). RF11.2 Identify, given a function f(x), values of x for which y = f (1x ) will have vertical asymptotes; and describe their relationship to the nonpermissible values of the related rational expression. RF11.3 Graph, with or without 1 technology, y = f ( x ) , given y = f (x) as a function or a graph, and explain the strategies used. Students will compare the graphs of a function and its reciprocal. Examples will include both linear and quadratic functions. When graphing a function and its reciprocal, there are invariant points that remain unchanged. Consider showing students the following table to highlight the invariant points using a function such as f (x) = 2x + 1. x f(x) 1 f (x ) 1 3 1 3 0.5 2 1 2 0 -0.5 -1 1 0 -1 1 undefined -1 Students should notice the invariant points can be located by setting f (x) = ±1. These points will help students when they graph reciprocal functions, as these will be common to both functions and are therefore the points of intersection. As students explore linear and quadratic reciprocal functions, there are certain characteristics that should be addressed in a class discussion. Ask students to graph f (x) = 2x + 1. f(x) f(x) = 2x + 1 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 -2 -3 -4 -5 182 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Presentation Authorized Resource • Pre-Calculus 11 Ask students to use graphing technology to analyse and describe the characteristics of a function and its reciprocal, and vice versa. As teachers observe this activity they should ask questions such as: Where do the graphs intersect? What do the vertical asymptotes of the reciprocal function correspond to in the original function? (RF11.1, RF11.2, RF11.3, RF11.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 7.4 Reciprocal Functions SB: pp.392-409 TR: pp.268-276 BLM: 7-3, 7-7 183 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF11 Continued... Achievement Indicators: RF11.1, RF11.2 Students should consider the following: RF11.3 Continued • If the point (x, y) is on the graph y = f (x), what do you notice about the point on the graph of the reciprocal function? • Is the sign of the reciprocal function the same as the sign of the original function? • What do you notice about the x-intercept of the linear function, the non-permissible value of the reciprocal function, and the location of the vertical asymptote? • What is the horizontal asymptote? • What are the invariant points? • Why do the intervals of decrease on the original function become intervals of increase on the reciprocal function? • Why do the intervals of increase on the original function become intervals of decrease on the reciprocal function? • What do you notice about the behaviour of the reciprocal function as it approaches the asymptotes? They should also recognize, from the table and the graph, that the x-intercept of the function produces a non-permissible value for the reciprocal function. This is the location of the vertical asymptote. When analyzing and comparing the graphs of the function and its reciprocal, it is important to note that as the y-values of one function increase the y-values of the other decrease and vice versa. This could be demonstrated by looking at parts of the graph on either side of the invariant points. As students recognize the shape of the graph along with the intercepts and the asymptotes, they should be able to sketch the graph of the reciprocal function. These ideas will now be extended to the graphs of quadratic functions and their reciprocals. There are three cases that need to be considered and explored with students, namely, quadratic functions that have one, two or no x-intercepts. 184 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Provide the following table with a set of statements. Ask students to describe the reasoning they used to decide whether each statement is true or false. Consider the function f (x ) = 1 True False Why I (we) think so 7.4 Reciprocal Functions SB: pp.392-409 TR: pp.268-276 BLM: 7-3, 7-7 3x − 4 (1) There is a vertical asymptote at x = 43 (2) There is a horizontal asymptote at y = 0. (3) f (0) = 1 4 (4) The point (4, 8) is on the graph of y = 3x -4 and the point (4, -8) is on the graph of y = 3x1−4 . (RF11.1, RF11.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 185 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF11 Continued... Achievement Indicators: First investigate a quadratic function with one x-intercept. Consider the graph of y =(x - 2)2 and its reciprocal y = ( x −12 )2 . RF11.1, RF11.2 y 6 5 4 3 2 1 RF11.3 Continued -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 1 2 3 4 5 6 x Using the graph as a visual aid, students should observe the following: • There is one x-intercept on the graph of y = f (x) located at x = 2. This corresponds to the location of the vertical asymptote on the graph of y = f (1x ) . • The graphs of y = f (x) and y = f (1x ) intersect where f (x) = ±1(in this case only at f (x) = 1). These are the invariant points. • The graph of y = f (1x ) is also asymptotic to the x-axis. The horizontal asymptote is located at y = 0. Similarly, students can investigate a quadratic function with two x-intercepts. Consider the graph of y = (x - 3)(x + 2) and it’s reciprocal y = ( x − 3)(1 x + 2 ) . y 10 9 8 7 6 5 4 3 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7 -8 -9 -10 -11 1 2 3 4 5 6 7 8 9 10 x Students should note the following features: • The x-intercepts of the graph of y = f (x), x = -2 and x = 3 correspond to the location of the vertical asymptotes on the graph of y = f (1x ) . 186 1 f ( x ) intersect • The graphs of y = f (x) and y = are the invariant points. • The equation of the horizontal asymptote is located at y = 0. • The parts of the graph of y = f (x) that were positive remain positive on the graph of y = f (1x ) . The parts of the graph of y = f (x) that were negative remain negative on the graph of y = f (1x ) . at f (x) = ±1. These ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Observation Authorized Resource • Pre-Calculus 11 7.4 Reciprocal Functions Students can work in pairs for this activity. Give one student the graph of a quadratic function and its reciprocal on the same set of axes. Ask him/her to turn to their partner and describe, using the characteristics of the quadratic function and its reciprocal, the graph they see. The other student will draw the graph based on the description from the student. Both students will then check to see if their graphs match. SB: pp.392-409 TR: pp.268-276 BLM: 7-3, 7-7 (RF11.1, RF11.2, RF11.3) Paper and Pencil • Provide the following table with a set of statements. Ask students to describe the reasoning they used to decide whether each statement is true or false. Consider the function f (x ) = 1 True False Why I (we) think so x 2 − 5x + 6 (1) There is a vertical asymptote at x = 2 (2) There is a vertical asymptote at x = -3 (3) f (0) = 1 6 (4) y = f (1x ) has x-intercepts at 2 and 3 (RF11.1, RF11.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 187 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF11 Continued... Achievement Indicators: RF11.1, RF11.2 RF11.3 Continued When graphing a quadratic function with no x-intercepts and its reciprocal, students should notice that the characteristics are different from the previous cases. Consider the graph of y = x2 + 3 and y = ( x 21+ 3) . y 5 4 3 2 1 -5 -4 -3 -2 -1 1 2 3 4 5 x -1 • The graph of y = f (x) has no x-intercepts. Therefore, the graph of y = f (1x ) has no vertical asymptotes. • The graph of y = f (x) does not have any points at f (x) = ±1. Therefore, there are no points of intersection between the graphs (i.e., there are no invariant points). • The equation of the horizontal asymptote is located at y = 0. This investigation helps students recognize the shape of the graph of a reciprocal function. Using the intercepts and asymptotes, students can proceed to graph a reciprocal function. RF11.4 Graph, with or without technology y= f(x) given y = f (1x ) as a function or a graph, and explain the strategies used. 188 Students will work backwards and sketch the graph of the original function given the graph of the reciprocal function. They should be able to determine by inspection whether the original function is linear or quadratic by analyzing the reciprocal function. It is important to remind students of the key concepts when working with reciprocal functions. The vertical asymptote of the reciprocal function is the x-intercept(s) of the original graph. The point (x,y) on the reciprocal function becomes (x, 1y ) on the original function. The two functions will intersect when f (x) = ±1. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Provide students with a reciprocal function. Ask them to sketch the graph of the original function, y = f (x), and explain the strategies they used. (RF11.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 7.4 Reciprocal Functions SB: pp.392-409 TR: pp.268-276 BLM: 7-3, 7-7 189 ABSOLUTE VALUE AND RECIPROCAL FUNCTIONS 190 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Systems of Equations Suggested Time: 8 Hours SYSTEMS OF EQUATIONS Unit Overview Focus and Context Outcomes Framework In this unit, students are introduced to linear-quadratic and quadraticquadratic systems of equations. The solutions to these systems are determined graphically by finding points of intersection. Students will solve the system of equations algebraically using the methods of substitution and elimination. They will then solve application problems involving a system of equations and verify their solutions. GCO Develop algebraic and graphical reasoning through the study of relations. SCO RF6 Solve, algebraically and graphically, problems that involve systems of linear-quadratic and quadraticquadratic equations in two variables. 192 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SYSTEMS OF EQUATIONS Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Mathematics 2200 Relations and Functions Relations and Functions RF9 Solve problems that involve systems of linear equations in two variables, graphically and algebraically. [CN, PS, R, T, V] Mathematics 3200 RF6 Solve, algebraically and graphically, problems that involve systems of linearquadratic and quadraticquadratic equations in two variables. [CN, PS, R, T, V] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 193 SYSTEMS OF EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF6 Solve, algebraically and graphically, problems that involve systems of linear-quadratic and quadratic-quadratic equations in two variables. [CN, PS, R, T, V] In Mathematics 1201, students modelled and solved systems of linear equations in two variables, graphically and algebraically (RF9). They created a linear system to model a situation, as well as wrote a description of a situation that might be modeled by a given linear system. They solved linear systems graphically, with and without technology, and progressed to solving linear systems symbolically using substitution and elimination. In this unit, students’ work will be extended to include systems of linearquadratic and quadratic-quadratic equations. Achievement Indicators: RF6.1 Explain the meaning of the points of intersection of a system of linear-quadratic or quadratic-quadratic equations. RF6.2 Explain, using examples, why a system of linear-quadratic or quadratic-quadratic equations may have zero, one, two or an infinite number of solutions. In Mathematics 1201, students created graphs of linear functions (RF6). They also investigated the meaning of the point of intersection of a system of linear equations (RF9). Students discovered that the intersection points of the graphs represented the solution to the system of equations. Ask students to sketch a line and a parabola on the same axis. As they examine their intersection points, students should recognize there are three situations possible for a linear-quadratic system. It can have zero, one or two solutions. y y x y x x Introducing the concepts of a tangent and a secant line may be waranted here. If a line intersects the parabola more than once, the line is referred to as a secant line. If the line intersects the parabola at exactly one point, the line is called a tangent line. Although this is not an outcome in this course, exposure to the terms will help them with courses to follow. Ask students to repeat the same activity by drawing two parabolas. Ask them how many solutions are possible for a quadratic-quadratic system. Students should recognize the same results with one exception; the system can also have an infinite number of solutions (i.e., coincident). 194 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SYSTEMS OF EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil Pre-Calculus 11 • Ask students to answer the following questions: Consider the parabola C whose equation is y = x2 - 2x - 3. (i) What is a possible equation for line L such that C and L form a system with no solution? one solution? two solutions? (ii) What is a possible equation for quadratic Q such that C and Q form a system with no solution? One solution? Two solutions? Infinite number of solutions? (RF6.1, RF6.2) • Ask students how they can tell by observation which of the following systems has no solution and which one has an infinite number of solutions. (i) −2 x 2 + 3x − y + 4 = 0 ⎫⎪ ⎬ −4 x 2 + 6x − 2 y + 8 = 0 ⎪⎭ (ii) y − 5 = 0 ⎫⎪ ⎬ y = −( x + 1) − 3⎪⎭ 8.1 Solving Systems of Equations Graphically Student Book (SB): pp.424-439 Teacher Resource (TR): pp.291299 BlackLine Master (BLM): 8-3, 8-4 2 (RF6.1, RF6.2) Performance • Ask students if the following lines intersect the parabola shown below at zero, one or two points. y (i) y = 21 x + 3 (ii) y = 21 x − 4 (iii) y = -2x (iv) x - 3 = 0 10 8 6 4 2 -2 -2 -4 2 4 6 8 x (RF6.1, RF6.2) Journal • Your friend stated that a system of linear-quadratic equations cannot have an infinite number of solutions whereas a system of quadratic-quadratic equations can. Ask students if they agree or disagree with this statement. They should justify their reasoning. (RF6.1, RF6.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 195 SYSTEMS OF EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF6 Continued ... Achievement Indicators: RF6.3 Determine and verify the solution of a system of linearquadratic or quadratic-quadratic equations graphically, with and without technology. Students will be working with the graphs of linear-quadratic and quadratic-quadratic systems to find their solutions graphically. Students will need to revist the methods used for graphing linear and quadratic relations. In Mathematics 1201, students were exposed to graphing linear functions using the slope-intercept method, slopepoint method, and using the x and y-intercept method. In this course, students explored the characteristics of standard form and vertex form of a quadratic function. Identifying the form of the equation will help students decide which method they should choose when graphing the linear or quadratic function. Graphing technology can also be used to solve a system of equations. Students can be exposed to, but not limited to a graphing calculator or graphing Apps. When students identify the points of intersection, remind them to verify the solution for both equations. There are limitations to solving a system by graphing. Non-integral intersection points are possible where students will have to estimate the coordinates. In such cases, an algebraic method of solving these systems is more efficient. RF6.4 Determine and verify the solution of a system of linearquadratic or quadratic-quadratic equations algebraically. Through their work with solving systems graphically, students should realize that algebra provides a more efficient means of finding the points of intersection. In Mathematics 1201, students were exposed to algebraic methods of substitution and elimination to solve a linear system. They will apply these methods to systems involving quadratic equations, including ones with rational coefficients. Provide students an opportunity to decide which algebraic method is more efficient when solving a system by focusing on the coefficients of like variables. If necessary, ask students to rearrange the equations so that like variables appear in the same position in both equations. When students solve the system using substitution or elimination, they will have to use factoring or the quadratic formula to solve the resulting quadratic equation. After determining the solution to the system, ensure that students verify the ordered pair satisfies the equations. Discuss with them why it is important to verify the solution in both equations. 196 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SYSTEMS OF EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Sam solved the system: 2 x − y = 9 ⎫⎪ ⎬ y = x 2 − 4x ⎪ ⎭ His solution was (3,-3). Ask students to verify whether his solution is correct. Ask them to explain how Sam’s results can be illustrated on a graph. (RF6.3) • 8.1 Solving Systems of Equations Graphically SB: pp.424-439 TR: pp.291-299 BLM: 8-3, 8-4 ⎧ x 2 − x + y = −2 Andrew was asked to solve the system: ⎪⎨ 2 ⎪⎩2 x − 4 x + 3 y = 0 The beginning of his solution is shown below. Ask students to finish the solution and verify their answer. Identify and correct any error(s) you or Andrew may have made. Andrew’s Solution: Multiply the first equation by -3 and add the second equation. x − x + y = −2 ⎯⎯→ −3 x + 3 x − 3 y = −2 × ( −3) 2 2 x − 4 x + 3 y = 0 ⎯⎯ → 2 2 2 −x − x 8.2 Solving Systems of Equations Algebraically 2 2x − 4x + 3y = 0 =−2 Now, solve -x2 - x = -2. (RF6.4) SB: pp.440-456 TR: pp.300-306 BLM: 8-3, 8-5 Observation • Ask students to participate in the following activity: Cards are distributed amongst the students. One third of the cards contain the coordinates of the points of intersection of two graphs. Another third of the cards contain a system of linear-quadratic or quadratic-quadratic equations written in slope y-intercept and vertex form. The final third contain the corresponding graphs for the system of equations. Students move around the classroom attempting to form a group of three by matching the cards containing the corresponding systems, graphs and points of intersections. (R) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 197 SYSTEMS OF EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF6 Continued ... Achievement Indicators: RF6.4 Continued Encourage students to predict the number of solutions before they solve the system graphically or algebraically. The discriminant, b2 - 4ac, can be used to help them with their prediction. Consider the system: y = 3x + 5 and y = 3x2 - 2x - 4. Ask students to equate the equations and simplify. Their resulting quadratic equation is 3x2 - 5x - 9 = 0. They determine the value of the discriminant to be 133. Students should realize that when the discriminant is greater than zero, there are two x values indicating two different solutions. This means that the linearquadratic system has two points of intersection. If the discriminant is equal to zero, there is one solution and a negative discriminant implies no point of interesection. They would then proceed to solve the system and the resulting quadratic equation using methods learned earlier in the Quadratic Equations unit (factoring, quadratic formula, completing the square). Remind students that a system of quadratic-quadratic equations can also produce an infinite number of solutions. When the functions are coincident (i.e., identical) students will recognize this results in a system where the left hand side equals the right hand side (i.e., 0 = 0). RF6.5 Model a situation, using a system of linear-quadratic or quadratic-quadratic equations. RF6.6 Relate a system of linearquadratic or quadratic-quadratic equations to the context of a given problem. 198 Students will model situations using a system of linear-quadratic and quadratic-quadratic equations connected to a variety of real-life contexts. When they can relate their learning to real-life applications it has more meaning. Paths of thrown or falling objects can be used to model both quadratic and linear functions. Ensure students understand and define the variables that are being used to represent the unknown quantities. Discuss with students that in order to solve a system, the number of unknowns must match the number of equations. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SYSTEMS OF EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Authorized Resource Paper and Pencil • Pre-Calculus 11 Ask students to create and solve the system of equations represented by the graphs below. They should verify that the solutions obtained algebraically match those found graphically. y y 6 4 4 2 2 -2 SB: pp.440-456 TR: pp.300-306 BLM: 8-3, 8-5 6 8 -4 8.2 Solving Systems of Equations Algebraically -2 2 -2 4 6 8 2 4 6 x -2 x -4 -4 -6 (RF6.3, RF6.4) • The price C, in dollars per share, of a high-tech stock has fluctuated over a twelve-year period and is represented by the parabola shown. The price C, in dollars per share, of a second high-tech stock has shown a steady increase during the same time period. y 60 50 40 30 20 10 -4 -2 (0, 14) 2 4 6 8 10 12 14 x -10 Ask students to answer the following: (i) Determine the system of equations that models the price over time. (ii) Solve the system. (iii) Determine the values where the two prices are the same. (RF6.5, RF6.6) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 199 SYSTEMS OF EQUATIONS Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF6 Continued... Achievement Indicator: RF6.7 Solve a problem that involves a system of linearquadratic or quadratic-quadratic equations, and explain the strategy used. 200 Students will solve problems involving a system of linear-quadratic or quadratic-quadratic equations. Students should decide which algebraic method is more efficient when solving the system of equations and they should be able to explain why they chose the preferred method. Explaining the meaning of a solution in a particular context and verifying the solution is an important part of solving systems. An answer to a system of equations may not necessarily be a possible solution in the context of the situation (i.e., inadmissible root). While solving a system of equations, a student may determine that the solutions are, for example, (-2, 5) and (5, 8). In the context of the problem the x-coordinate may represent a measurement such as time or length. The point (-2, 5), therefore, would have to be rejected as a solution leading to only one possible answer. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 SYSTEMS OF EQUATIONS General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Ask students to solve the following: Pre-Calculus 11 (i) A rectangular field has a perimeter of 500 m and an area of 14 400 m2. Find the length of the sides. (RF6.7) 8.2 Solving Systems of Equations Algebraically (ii) Ask students to write a system of equations to represent two numbers that differ by 4 and whose squares have a sum of 136? (RF6.7) SB: pp.440-456 TR: pp.300-306 BLM: 8-3, 8-5 (iii) A right triangle has a hypotenuse 10 cm long. If the perimeter is 22 cm, find the lengths of the other two sides. (RF6.7) (iv) A sky diver jumped from a tower and fell freely for several seconds before releasing her parachute. Her height, h, in metres, above the ground at any time is given by: h = -4.9t2 + 5000 before she released her parachute, and h = -4t + 4000 after she released her parachute. If t represents time in seconds, how long after jumping did she release her parachute? How high was she above the ground at that time? (RF6.7) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 201 SYSTEMS OF EQUATIONS 202 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 Linear and Quadratic Inequalities Suggested Time: 8 Hours LINEAR AND QUADRATIC INEQUALITIES Unit Overview Focus and Context Outcomes Framework In this unit, students will solve quadratic inequalities in one variable as well as linear and quadratic inequalities in two variables. They will investigate finding the solutions using various methods, ideally leading to a preferred personal strategy. They will apply the various skills developed to solve a number of real-world problems. GCO Develop algebraic and graphical reasoning through the study of relations. SCO RF7 Solve problems that involve linear and quadratic inequalities in two variables. SCO RF8 Solve problems that involve quadratic inequalities in one variable. 204 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 LINEAR AND QUADRATIC INEQUALITIES Process Standards Key [C] Communication [CN] Connections [ME] Mental Mathematics and Estimation [PS] [R] [T] [V] Problem Solving Reasoning Technology Visualization SCO Continuum Mathematics 1201 Mathematics 2200 Mathematics 3200 Relations and Functions Relations and Functions Relations and Functions RF6 Relate linear equations expressed in: • slope-intercept form (y = mx + b) • general form (Ax +By+C=0) • slope-point form (y-y1=m(x-x1)) to their graphs [CN, R, T, V] RF7 Solve problems that involve linear and quadratic inequalities in two variables. RF12 Graph and analyze polynomial functions (limited to polynomial functions of degree ≤ 5). [C, PS, T, V] [C, CN, T, V] RF8 Solve problems that involve quadratic inequalities in one variable. [CN, PS, V] ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 205 LINEAR AND QUADRATIC INEQUALITIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF7 Solve problems that involve linear and quadratic inequalities in two variables. [C, PS, T, V] In Grade 9, students solved single variable linear inequalities with rational coefficients and graphed their solutions on a number line (9PR4). Students are familiar with the inequality symbols, the terms continuous and discrete data, as well as their effect on a graph. In Mathematics 1201, students graphed linear relations expressed in slopeintercept form, general form and point-slope form (RF6). Students are familiar with verifying solutions to linear equations and linear inequalities using substitution. They also verified solutions by examining the number line graphs of inequalities. Students have been exposed to the concept that the solution of a linear inequality consists of a set of points while the solution of a linear equation has only one solution. In this unit, students will solve quadratic inequalities in one variable, as well as linear and quadratic inequalities in two variables. Achievement Indicators: RF7.1 Explain, using examples, how test points can be used to determine the solution region that satisfies a linear inequality. RF7.2 Explain, using examples, when a solid or broken line should be used in the solution for a linear inequality. RF7.3 Sketch, with or without technology, the graph of a linear inequality. 206 Students will first investigate linear inequalities with two variables. The solution of a linear inequality in two variables, such as Ax + By > C, consists of any ordered pair (x, y) that produces a true statement when the values of x and y are substituted into the inequality. Since there may be infinitely many solutions, it is not possible for students to list all of them. Discussion around the difference between an equation and an inequality should highlight that the solution to an inequality in two variables is represented by a region. Explain to students that the boundary line Ax + By = C divides the coordinate plane into two parts, where one part represents the solutions to the inequality. To be successful with graphing inequalities in two variables, students must be proficient with graphing linear equations, with and without technology. Identifying the form of the equation will help students decide which method they should choose when graphing the line (i.e., slope-intercept, point-slope or x-and y-intercept method). Initially, students use test points to investigate the region of the plane that satisfies the inequality which will help guide them to an understanding of when to shade above or below the boundary. Working through several examples should help them make the connection between the sign of the inequality, the shaded region and whether to use a solid or broken line. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 LINEAR AND QUADRATIC INEQUALTITES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Ask students to complete a table similar to the one below: Inequality shade above or below broken or solid line y ≤ -2x + 8 x+y>2 (RF7.2, RF7.3) Pre-Calculus 11 • A student was asked to graph 3x - 2y > 12. His solution is shown below. Ask stidents to identify and correct his error(s). Student Solution: 3x - 2y > 12 -2y > -3x + 12 y 10 8 6 4 y > 23 x − 6 2 -10 -8 -6 -4 -2 2 4 6 8 10 x -2 9.1 Linear Inequalities in Two Variables -4 -6 -8 -10 (RF7.3) Performance • Ask students to work in groups of two for this activity. Each group should be given a deck of cards consisting of 12 with a graph, 12 with an inequality and one “YOU WIN” card. Student Book (SB): pp.464-475 Teacher Resource (TR): pp.317323 BlackLine Master (BLM): 9-3, 9-4 A dealer is chosen to deal the cards to each student. Students make pairs consisting of a graph and its matching inequality. These pairs are set aside face up on the table. Students take turns selecting a card from their partner and making matches if possible. Play continues until all matches are made. The students who has the YOU WIN card at the end is declared the winner. (RF7.3) Journal • Your friend asks you to explain the difference between graphing 3y + 2x = 4, 3y + 2x > 4, and 3y + 2x ≥ 4. Ask students to write a response. (RF7.2) • The solution region for the inequality 5x - 3y > 10 is above the line since it contains a “greater than sign”. Ask students if they agree or disagree with their statement. They should explain their reasoning. (RF7.3) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 207 LINEAR AND QUADRATIC INEQUALITIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF7 Continued ... Achievement Indicators: RF7.1, RF7.2, RF7.3 Continued When solving single variable inequalities, students sometimes forget to reverse the sign of the inequality when multiplying or dividing both sides by a negative number. A brief review may be necessary as this error will also affect the feasible region when solving inequalities in two variables. Students should be provided with opportunities to write the equation of the linear inequality given its graph. They will find the equation of the boundary line and then use the given shaded region to determine the correct inequality. Remind students what a broken or solid boundary line represents. RF7.4 Solve a problem that involves a linear inequality. Students will relate inequalities to real life contexts. Problems that can be expressed as an inequality in two variables require students to find two unknown quantities under certain constraints. Students will translate the word problem into an inequality. Ensure that they understand and define the variables that are being used to represent the unknown quantities. They should also realize that the shaded feasible region represents all possible combinations for the two quantities. An example such as the following could be used to highlight feasible and realistic solutions to a problem. • With two minutes left in a basketball game, your team is 12 points behind. What are two different numbers of 2-point and 3-point shots your team could score to earn at least 12 points? Students should rcognize that the domain and range contain only positive values. Ask them to identify realistic points in the feasible region considering the time remaining in the game. 208 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 LINEAR AND QUADRATIC INEQUALTITES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to answer the following: (i) Susan plans to spend a maximum of 15 hours reviewing Math and Biology in preparation for examinations. Draw a graph showing how much time she could spend studying each subject. (RF7.4) (ii) A contractor has at least one hundred tonnes of soil to be moved using two trucks. One truck has a 4 ton capacity and the other has a 5 ton capacity. Make a graph to show the various combinations of loads the two trucks could carry to complete the job. (RF7.4) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 9.1 Linear Inequalities in Two Variables SB: pp.464-475 TR: pp.317-323 BLM: 9-3, 9-4 209 LINEAR AND QUADRATIC INEQUALITIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF8 Solve problems that involve quadratic inequalities in one variable. [CN, PS, V] Achievement Indicators: RF8.1 Determine the solution of a quadratic inequality in one variable, using strategies such as case analysis, graphing, roots and test points, or sign analysis; and explain the strategy used. RF8.2 Represent and solve a problem that involves a quadratic inequality in one variable. RF8.3 Interpret the solution to a problem that involves a quadratic inequality in one variable. In Mathematics 1201, students were introduced to domain and range (RF1). They used interval notation and set notation to represent solution sets. This will now be extended to solving quadratic inequalities. It is important to have a discussion with students about what it means to solve a quadratic inequality in one variable. When solving ax2 + bx + c ≥ 0, for example, students should interpret this as finding the possible x-values where the corresponding y-values are zero or positive. In other words, when is the graph of y = ax2 + bx + c on or above the x-axis. This can be done graphically or algebraically. Students may have a better understanding of the algebraic method, however, if it is related back to the graph. Regardless of the technique used, proficiency in solving quadratic equations is important. One strategy that can be used to solve a quadratic inequality involves roots and test points. Solving an inequality such as x2 + 2x - 3 < 0, students first determine the roots of the quadratic equation x2 + 2x - 3 = 0. They will then use a sign diagram consisting of a number line and test points to determine the intervals that satisfy the inequality. The sign diagram was introduced in the Absolute Value and Reciprocal Unit. y y = (x – 1)(x + 3) 9 6 3 -5 -4 -3 -2 -1 1 2 3 4 5 x -3 (-1, -4) -6 -9 -12 y-values - y-values + test point x = -4 -3 test point x=0 y-values + 1 test point x=2 x Relating the sign diagram to the x-axis of the graph, students use test points to determine if the function is positive or negative. They should recognize the roots -3 and 1 are not part of the solution. Students can write the solution set as set notation, {x/ -3 < x < 1, x ∈ \ } or interval notation (-3, 1). Students will solve problems that involve a quadratic inequality in one variable. These types of problems will be similar to solving a quadratic equation except there is usually a minimum or maximum constraint. Students should relate their solution to the context of the problem. 210 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 LINEAR AND QUADRATIC INEQUALTITES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Ask students to answer the following: Pre-Calculus 11 (i) Given A = x2 - 7 and B = -4x + 5, for what values of x is A < B? (RF8.1) 9.2 Quadratic Inequalties in One Variable (ii) Use the graphs shown below to state the x-values for which f (x) > 0. y 8 7 6 5 4 3 2 1 -6 -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7 -8 -9 y y = f(x) 1 2 3 4 5 6 SB: pp.476-487 TR: pp.324-329 BLM: 9-3, 9-5 8 7 6 5 4 3 2 1 x -5 -4 -3 -2 -1-1 -2 -3 -4 -5 -6 -7 -8 -9 1 2 3 4 5 6 x y = f(x) (RF8.1) (iii) When a projectile is fired into the air, its height h, in metres, t seconds later is given by the equation h = 11t - 3t2. When is the projectile at least 6 m above the ground? (RF8.2, RF8.3) (iv) When a baseball is hit by a batter, the height of the ball, h(t), at time t, is determined by the equation h(t) = -16t2 + 64t + 4. For which interval of time is the height of the ball greater than or equal to 52 feet? (RF8.2, RF8.3) (v) The surface area, A, of a cylinder with radius r is given by the formula A = 2r2 - 5r. What possible radii would result in an area that is greater than 12 cm2? (RF8.2, RF8.3) Journal • Your friend asks you to explain the difference between solving x2 - 2x - 3 = 0, x2 - 2x - 3 ≥ 0, and x2 - 2x - 3 < 0. Ask students to write a response. (RF8.1, RF8.2) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 211 LINEAR AND QUADRATIC INEQUALITIES Relations and Functions Outcomes Elaborations—Strategies for Learning and Teaching Students will be expected to RF7 Continued... Achievement Indicators: RF7.5 Explain, using examples, how test points can be used to determine the solution region that satisfies a quadratic inequality. RF7.6 Explain, using examples, when a solid or broken line should be used in the solution for a quadratic inequality. RF7.7 Sketch, with or without technology, the graph of a quadratic inequality. RF7.8 Solve a problem that involves a quadratic inequality. 212 The strategies used to graph linear inequalities in two variables will now be extended to quadratic inequalities in two variables. Solving quadratic inequalities in one variable involved students finding the set of all x values that made a particular inequality true. Solving a quadratic inequality in two variables implies finding, graphically, all the coordinate pairs (x, y) that make a particular inequality true. The graph of a quadratic function separates the plane into two regions, one of which contains all the points that satisfy the inequality. Students will graph the corresponding function that is associated with the inequality and use test points to see which region should be shaded. They should continue to ask themselves if the graph should be drawn with a solid or broken curve. Students will also write an inequality to describe a graph, given the function defining its boundary. Ensure students recognize whether the parabola is a solid or broken curve. Ask them how to identify the inequality using test points. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 LINEAR AND QUADRATIC INEQUALTITES General Outcome: Develop algebraic and graphical reasoning through the study of relations. Suggested Assessment Strategies Resources/Notes Paper and Pencil Authorized Resource • Pre-Calculus 11 Ask students to sketch the feasible region for each of the following inequalities. (i) y ≥ x2 + 2x - 3 2 (ii) y < −2( x − 1) + 4 (iii) y < 13 ( x − 3)( x + 3) 9.3 Quadratic Inequalities in Two Variables (RF7.7) • Give students a number of linear and quadratic inequalities such as: SB: pp.488-500 TR: pp.330-336 BLM: 9-3, 9-6 (i) 2x - 3y < 7 (ii) -x +4y ≥ -6 (iii) y > -2(x + 1)2 - 4 (iv) y ≤ -x2 + 1 Students can work in groups. Provide them with a list of 5 points and have them work together to determine which inequality, if any, satisfies their point. (RF7.1, RF7.5) • The base of a rectangular bin currently has dimensions 12 m by 5 m. The base is to be enlarged by an equal amount on the width and length so that the area is more than doubled. Ask students by how much should the length and width be increased to produce the desired area? x 12 5 x (RF7.8) • Ask students to determine an inequality to match each graph. y y 6 6 5 5 4 4 3 3 2 2 1 1 -8 -7 -6 -5 -4 -3 -2 -1 1 -1 -2 -3 -4 -5 -6 Inequality: 2 3 4 x -2 -1 1 2 3 4 5 6 7 8 9 10 x -1 -2 -3 -4 -5 -6 Inequality: (RF7.6) ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 213 LINEAR AND QUADRATIC INEQUALITIES 214 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 REFERENCES REFERENCES American Association for the Advancement of Science [AAAS–Benchmarks]. Benchmarks for Science Literacy. New York, NY: Oxford University Press, 1993. Armstrong, Thomas. 7 Kinds of Smart: Identifying and Developing Your Many Intelligences. New York, NY: Plume, 1993. British Columbia Ministry of Education. The Primary Program: A Framework for Teaching. Victoria, BC: British Columbia Ministry of Education, 2000. Brown, Richard and David Robbins. Advanced Mathematics - A Precalculus Course. Markham, Ontario: Houghton Mifflin Canada Limited., 1989. Caine, Renate Nummela and Geoffrey Caine. Making Connections: Teaching and the Human Brain. Alexandria, VA: Association for Supervision and Curriculum Development, 1991. Government of Newfoundland and Labrador. Foundations for the Atlantic Canada Mathematics Curriculum. Halifax, NS: Atlantic Provinces Education Foundation. Government of Newfoundland and Labrador: Department of Education. Mathematics Grade 10 Interim Edition Curriculum Guide 2011. Retrieved March 2012 from the World Wide Web: http://www.ed.gov. nl.ca/edu/k12/curriculum/guides/mathematics/index.html. Hope, Jack A. et al. Mental Math in Junior High. Palo Alto, CA: Dale Seymour Publications, 1988. Hope, Jack A. et al. Mental Math in the Primary Grades. Palo Alto, CA: Dale Seymour Publications, 1988. Keeley, Page and Cheryl Rose Tobey. Mathematics Formative Assessment. Thousand Oaks, California: Corwin, 2011. Kelly, Brendan. et al. Addison-Wesley Mathematics 12. Don Mills, Ontario: Addison-Wesley Publishers Limited., 1991. McAskill, Bruce. et al. Pre-Calculus 11. Toronto, ON: McGraw-Hill Ryerson Limited., 2011. ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012 215 REFERENCES National Council of Teachers of Mathematics. Computation, Calculators, and Common Sense: A Position of the National Council of Teachers of Mathematics. May 2005. http://www.nctm.org/about/pdfs/position/ computation.pdf (Accessed February 22, 2007). Rubenstein, Rheta N. “Mental Mathematics beyond the Middle School: Why? What? How?” Mathematics Teacher 94, 6 (September 2001), pp. 442–446. Shaw, J. M. and M. J. P. Cliatt. “Developing Measurement Sense.” In P. R. Trafton (ed.), New Directions for Elementary School Mathematics: 1989 Yearbook (Reston, VA: National Council of Teachers of Mathematics, 1989), pp. 149–155. Small, Marian. Making Math Meaningful to Canadian Students, K-8. Toronto, ON: Nelson Education, 2008. Steen, L. A., ed. On the Shoulders of Giants: New Approaches to Numeracy. Washington, DC: Mathematical Sciences Education Board, National Research Council, 1990. Travers, Kenneth. et al. Using Advanced Algebra. Toronto: Collier Macmillan Canada, Inc., 1985. Van de Walle, John A. Elementary and Middle School Mathematics: Teaching Developmemtally. 4th ed. Boston, MA: Addison Wesley LAyman, INC., 2001. Western and Northern Canadian Protocol for Collaboration in Basic Education (Kindergarten to Grade 12). The Common Curriculum Framework for K–9 Mathematics: Western and Northern Canadian Protocol – May 2006 and The Common Curriculum Framework for Grades 10–12 – January 2008. Reproduced (and/or adapted) by permission. All rights reserved. 216 ADVANCED MATHEMATICS 2200 CURRICULUM GUIDE - INTERIM 2012

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