|Calculus & Vectors|
|Prerequisite||MHF4U, Advanced Functions, Grade 12, University (may be taken concurrently)|
|Curriculum policy document:||Download|
|Course developer:||Peak Centre Online|
This course builds on students’ previous experience with functions and their developing understanding of rates of change. Students will solve problems involving geometric and algebraic representations of vectors and representations of lines and planes in three dimensional space; broaden their understanding of rates of change to include the derivatives of polynomial, sinusoidal, exponential, rational, and radical functions; and apply these concepts and skills to the modelling of real-world relationships. Students will also refine their use of the mathematical processes necessary for success in senior mathematics. This course is intended for students who choose to pursue careers in fields such as science, engineering, economics, and some areas of business, including those students who will be required to take a university-level calculus, linear algebra, or physics course.
Unit One: Vectors
There are four main topics pursued in this initial unit of the course. These topics are: an introduction to vectors and scalars, vector properties, vector operations and plane figure properties.
Unit Two: Linear Dependence and Coplanarity
In this unit, the addition, subtraction, and scalar multiplication of Cartesian vectors are all investigated. Students investigate the concepts of linear dependence and independence, and collinearity and coplanarity of vectors.
Unit Three: Vector Applications
In this unit, the properties of vector products are investigated and proven. These vector products will be revisited to predict characteristics of the solutions of systems of lines and planes in the intersections of lines and planes.
Unit Four: Intersection of Lines and Planes
This unit begins with students determining the vector, parametric and symmetric equations of lines in R2 and R3. Students will also determine the vector, parametric, symmetric and scalar equations of planes in 3-space. Students will also learn the intersections of lines in 3-space and the intersections of a line and a plane in 3-space. Students will learn to determine the intersections of two or three planes by setting up and solving a system of linear equations in three unknowns. Students will also define the terms related to matrices while adding, subtracting, and multiplying them. Students will solve systems of linear equations involving up to three unknowns, using row reduction of matrices, with and without the aid of technology and interpreting row reduction of matrices as the creation of new linear systems equivalent to the original constitute the final two new topics of this important unit.
Unit Five: Concepts of Calculus
In this unit students will deal with rates of change problems and the limit concept.. To further investigate the concept of a limit, the unit briefly looks at the relationship between a secant line and a tangent line to a curve. To this point in the course students have been given a fixed point and have been asked to find the tangent slope at that value, in this section of the unit students will determine a tangent slope function similar to what they had done with a secant slope function. Sketching the graph of a derivative function is the final skill and topic.
Unit Six: Derivatives
In this unit , students will examine the power rule, the product rule, the quotient rule and the chain rule followed by a study of the derivatives of composite functions. Next, students will find the derivative of relations that cannot be written explicitly in terms of one variable. Additionally, students will apply the rules they have already developed to find higher order derivatives.Unit
Unit Seven: Curve Sketching
In this course, the key features of a properly sketched curve are all reviewed separately before putting them all together into a full sketch of a curve.
Unit Eight: Derivative Applications and Related Rates
During this unit students will look at each of these types of problems individually: Pythagorean Theorem Problems (these include ladder and intersection problems), Volume Problems (these usually involve a 3-D shape being filled or emptied), Through Problems, Shadow problems and General Rate Problems.
Unit Nine: Derivative of Exponents and Log Functions-Exponential Functions
This unit begins with examples and exercises involving exponential and logarithmic functions using Euler’s number (e). Students will now look at how they can use their established rules to find the derivatives of such functions.
Unit Ten: Trig Differentiation and Application
In this unit students will review trigonometry, kicks off this unit and then study special angles and the CAST rule which has been developed to identify which of the basic trigonometric ratios is positive and negative in the four quadrants. Students will then solve trigonometric equations using the CAST rule to locate other solutions.
Final Exam 30%
This is a proctored exam worth 30% of your final grade
|Total hours||110 hours|
Teaching and Learning Strategies & Strategies for Assessment
Teaching and Learning Strategies & Strategies for Assessment
Students will follow a similar pattern of instructions in all units. To begin students will be involved in the exploration of an investigation of a concept. Then they will apply what they have learned in several real life scenarios or applications of the concept. Students will see solutions to applications after they try to solve them for themselves. Then students will complete assignments where no solutions are provided and submit these for assessment. Finally the unit ends with a test.
A wide variety of instructional strategies are used to provide learning opportunities to accommodate a variety of learning styles, interests and ability levels. Seven mathematical processes will form the heart of the teaching and learning strategies used:
- Communicating: To improve student success there will be several opportunities for students to share their understanding both in oral as well as written form.
- Problem-solving: Scaffolding of knowledge, detecting patterns, making and justifying conjectures, guiding students as they apply their chosen strategy, directing students to use multiple strategies to solve the same problem, when appropriate, recognizing, encouraging, and applauding perseverance, discussing the relative merits of different strategies for specific types of problems.
- Reasoning and proving: Asking questions that get students to hypothesize, providing students with one or more numerical examples that parallel these with the generalization and describing their thinking in more detail.
- Reflecting: Modeling the reflective process, asking students how they know.
- Selecting Tools and Computational Strategies: Modeling the use of tools and having students use technology to help solve problems.
- Connecting: Activating prior knowledge when introducing a new concept in order to make a smooth connection between previous learning and new concepts, and introducing skills in context to make connections between particular manipulations and problems that require them.
- Representing: Modeling various ways to demonstrate understanding, posing questions that require students to use different representations as they are working at each level of conceptual development – concrete, visual or symbolic, allowing individual students the time they need to solidify their understanding at each conceptual stage.
Strategies for Assessment and Evaluation of Student Performance
The teacher will obtain assessment information through a variety of means as indicated in the chart below. Assessment and Evaluation Strategies are to include the evidence or proof the teacher sees in the Product, Observations and Conversations related to the curriculum expectations. The student must demonstrate achievement of the course expectations. Once demonstrated, the student is assigned a level of achievement.
- Assessment for: takes place in preparation for course or unit learning.
- Assessment as: takes place during or while learning.
- Assessment of: takes place after learning.
These assessments and evaluations take place throughout the course.
Teaching & Learning Strategies
Students will analyze literary texts from contemporary and historical periods, interpret informational and graphic texts, and create oral, written, and media texts in a variety of forms. An important focus will be on the use of strategies that contribute to effective communication.
Teachers differentiate instruction to meet the diverse learning needs of students. Instructors use Discussion Boards, Google Apps for Education, Multi-Media element, constant valuable feedback, Google docs, Google forms, Google slides, Google Drive to meet the needs of students and to assist students in reflecting on their learning, and in setting goals for improvement in key areas while developing 21st-century skills. These tools help facilitate the development of 21st-century learners and ensure the development of students that can self-assess, work independently and demonstrate their ability to critically analyze text.
Identifying and developing skills and strategies
Students learn to choose and utilize varied techniques taught through video lessons, assignments, activities, and student exemplars to become effective readers, writers, and oral communicators.
Several opportunities are provided for students to write and communicate orally and for teachers to assess work based on conversation and observation.
Generating ideas and topics
Teachers encourage students to design their own approaches to the material by maintaining frequent (often daily) online communication with students, by allowing some freedom in how students respond to topics and questions, and by encouraging students’ independent thinking through discussion posts.
Various approaches to researching are practiced. Students learn how to use various online research tools, cite sources, evaluate web sources and provide works cited page at the end of longer assignments using MLA formatting.
Students learn to critically analyze texts and to infer through their deeper analysis. . Students use their critical thinking skills to identify themes, morals, and the use of literary elements and devices.
Producing published work and making presentations
Students engage in the editing and revising process, including self-revision, peer revision, and teacher revision all of which strengthen texts with the aim to publish or present student work.
Through the variety of assignments, lessons and discussions, students reflect on the learning process, focus on areas for improvement, and make world to text, self to text and text to text connections between course content and their personal experiences.
The Final Grade
The evaluation for this course is based on the student’s achievement of curriculum expectations and the demonstrated skills required for effective learning.
The percentage grade represents the quality of the student’s overall achievement of the expectations for the course and reflects the corresponding level of achievement as described in the achievement chart for the discipline.
A credit is granted and recorded for this course if the student’s grade is 50% or higher. The final grade for this course will be determined as follows:
- 70% of the grade will be based upon evaluations conducted throughout the course. This portion of the grade will reflect the student’s most consistent level of achievement throughout the course, although special consideration will be given to more recent evidence of achievement
- 10% of the grade will be based on a Rich summative task administered in the last weeks of the course. This RST will be based on an evaluation of achievement from all four categories of the Achievement Chart for the course and of expectations from all units of the course.
- 20% of the grade will be based on a final examination administered at the end of the course. This exam will be based on an evaluation of achievement from all four categories of the Achievement Chart for the course and of expectations from all units of the course. This exam includes well-formulated multiple-choice questions, long-answer type questions and an essay.
The Report Card
Student achievement will be communicated formally to students via an official report card. Report cards are issued at the midterm point in the course, as well as upon completion of the course.
Each report card will focus on two distinct, but related aspects of student achievement. First, the achievement of curriculum expectations is reported as a percentage grade. Additionally, the course median is reported as a percentage. The teacher will also provide written comments concerning the student’s strengths, areas for improvement, and next steps. Second, the learning skills are reported as a Needs Improvement, Satisfactory, Good and Excellent. The report card also indicates whether an OSSD credit has been earned. Upon completion of a course, Kanata Academy will send a copy of the report card back to the student’s home school (if in Ontario) where the course will be added to the ongoing list of courses on the student’s Ontario Student Transcript. The report card will also be sent to the student’s home address.
Program Planning Considerations
Teachers who are planning a program in the Arts must take into account considerations in a number of important areas. Essential information that pertains to all disciplines is provided in the companion piece to this document, The Ontario Curriculum, Grades 9 to 12: Program Planning and Assessment, 2000. The areas of concern to all teachers that are outlined here include the following:
- Types of secondary school courses
- Education for exceptional students
- The role of technology in the curriculum
- English as a second language (ESL) and English literacy development (ELD)
- Career education
- Cooperative education and other workplace experiences
- Health and safety
- Financial literacy
- Critical Thinking
- Environmental Education
Education for Exceptional Students
In planning courses, teachers should take into account the needs of exceptional students as set out in their Individual Education Plan. Art courses reflect the creative part of our world, which offers a vast array of opportunities for exceptional students. Students who use alternative techniques for communication may find a venue for their talents as artists. Just as Art responds to the creative demands of the greater world of work, Art courses are largely shaped by the needs and demands of students who will all eventually end up in this greater world.
The Role of Technology in the Curriculum
Information and communications technologies (ICT) provide a range of tools that can significantly extend and enrich teachers’ instructional strategies and support students’ language learning. ICT tools include multimedia resources, databases, Internet websites, digital cameras, and word-processing programs. Tools such as these can help students to collect, organize, and sort the data they gather and to write, edit, and present reports on their findings. Information and communications technologies can also be used to connect students to other schools, at home and abroad, and to bring the global community into the virtual classroom. Although the Internet is a powerful learning tool, there are potential risks attached to its use. All students must be made aware of issues of Internet privacy, safety, and responsible use, as well as of the potential for abuse of this technology, particularly when it is used to promote hatred. Information technology is considered a learning tool that must be accessed by students when the situation is appropriate. As a result, students will develop transferable skills through their experience with word processing, internet research, presentation software, and telecommunication tools.
English As a Second Language and English Literacy Development (ESL/ELD)
With exposure to the English language in a supportive learning environment, most young children will develop oral fluency quite quickly, making connections between concepts and skills acquired in their first language and similar concepts and skills presented in English. However, oral fluency is not a good indicator of a student’s knowledge of vocabulary or sentence structure, reading comprehension, or other aspects of language proficiency that play an important role in literacy development and academic success. Research has shown that it takes five to seven years for most English language learners to catch up to their English-speaking peers in their ability to use English for academic purposes. Moreover, the older the children are when they arrive, the greater the language knowledge and skills that they have to catch up on, and the more direct support they require from their teachers. Responsibility for students’ English-language development is shared by the course teacher, the ESL/ELD teacher (where available), and other school staff. Volunteers and peers may also be helpful in supporting English language learners in the language classroom. Teachers must adapt the instructional program in order to facilitate the success of these students in their classrooms.
Appropriate adaptations include:
- Modification of some or all of the subject expectations so that they are challenging but attainable for the learner at his or her present level of English proficiency, given the necessary support from the teacher;
- Use of a variety of instructional strategies (e.g., extensive use of visual cues, graphic organizers, scaffolding; previewing of textbooks, pre-teaching of key vocabulary; peer tutoring; strategic use of students’ first languages);
- Use of a variety of learning resources (e.g., visual material, simplified text, bilingual dictionaries, and materials that reflect cultural diversity);
- Use of assessment accommodations (e.g., granting of extra time; use of oral interviews, demonstrations or visual representations, or tasks requiring completion of graphic organizers or cloze sentences instead of essay questions and other assessment tasks that depend heavily on proficiency in English).
Note: When learning expectations in any course are modified for an English language learner (whether the student is enrolled in an ESL or ELD course or not), this information must be clearly indicated on the student’s report card.
As online students progress through online courses, teachers are available to help the student prepare for employment in a number of diverse areas. With the help of teachers, students will learn to set and achieve goals and will gain experience in making meaningful decisions concerning career choices. The skills, knowledge and creativity that students acquire through this online course are essential for a wide range of careers. Throughout their secondary school education, students will learn about the educational and career opportunities that are available to them; explore and evaluate a variety of those opportunities; relate what they learn in their courses to potential careers in a variety of fields; and learn to make appropriate educational and career choices.
Cooperative Education and Other Workplace Experiences
By applying the skills they have developed, students will readily connect their classroom learning to real-life activities in the world in which they live.Cooperative education and other workplace experiences will broaden their knowledge of employment opportunities in a wide range of fields. We will try to help students link to Ministry programs to ensure that students have information concerning programs and opportunities.
Financial literacy may be defined as having the knowledge and skills needed to make responsible economic and financial decisions with competence and confidence. Since making financial decisions has become an increasingly complex task in the modern world, students need to have knowledge in various areas and a wide range of skills in order to make informed decisions about financial matters. Students need to be aware of risks that accompany various financial choices. They need to develop an understanding of world economic forces as well as ways in which they themselves can respond to those influences and make informed choices. We consider it essential that financial literacy be considered an important attribute of a well-educated population. In addition to acquiring knowledge in such specific areas as saving, spending, borrowing, and investing, students need to develop skills in problem-solving, inquiry, decision making, critical thinking, and critical literacy related to financial and other issues. The goal is to help students acquire the knowledge and skills that will enable them to understand and respond to complex issues regarding their own personal finances and the finances of their families, as well as to develop an understanding of local and global effects of world economic forces and the social, environmental, and ethical implications of their own choices as consumers. We are working to embed financial literacy expectations and opportunities in all courses as appropriate, as part of the ongoing curriculum review process.
Critical thinking is the process of thinking about ideas or situations in order to understand them fully, identify their implications, make a judgment, and/or guide decision-making. Critical thinking includes skills such as questioning, predicting, analyzing, synthesizing, examining opinions, identifying values and issues, detecting bias, and distinguishing between alternatives. Students who are taught these skills become critical thinkers who can move beyond superficial conclusions to a deeper understanding of the issues they are examining. They are able to engage in an inquiry process in which they explore complex and multifaceted issues, and questions for which there may be no clear-cut answers.
Students use critical-thinking skills when they assess, analyze, and/or evaluate the impact of something and when they form an opinion about something and support that opinion with a rationale. In order to think critically, students need to examine the opinions and values of others, detect bias, look for implied meaning, and use the information gathered to form a personal opinion or stance, or a personal plan of action with regard to making a difference. Students approach critical thinking in various ways. Some students find it helpful to discuss their thinking, asking questions and exploring ideas. Other students, including many First Nations, Mêtis, and Inuit students, may take time to observe a situation or consider a text carefully before commenting; they may prefer not to ask questions or express their thoughts orally while they are thinking.
The development of these critical-thinking skills is supported in every course. As students work to achieve the curriculum expectations in their particular course, students frequently need to identify the possible implications of choices. As they gather information from a variety of sources, they need to be able to interpret what they are listening to, reading, or viewing; to look for instances of bias, and to determine why a source might express a particular bias.
Health and Safety
In order to provide a suitable learning environment for staff and students, it is critical that classroom practice and the learning environment complies with relevant federal, provincial, and municipal health and safety legislation and by-laws, including, but not limited to, the Workplace Safety and Insurance Act, the Workplace Hazardous Materials Information System (WHMIS), the Food and Drug Act, the Health Protection and Promotion Act, the Ontario Building Code, and the Occupational Health and Safety Act (OHSA). The OHSA requires all schools to provide safe and productive learning and work environments for both students and employees.
Helping students become environmentally responsible is a role the school takes seriously. The first goal is to promote learning about environmental issues and solutions. The second goal is to engage students in practicing and promoting environmental stewardship in their communities. The third goal stresses the importance of the education system providing leadership by implementing and promoting responsible environmental practices so that all stakeholders become dedicated to living more sustainably. Environmental education teaches students about how the planet’s physical and biological systems work, and how we can create a more sustainable future. Good curriculum design following the resource document – The Ontario Curriculum, Grades 9-12: Environmental Education, Scope and Sequence of Expectations, 2011, will assist staff to weave environmental education in and out of the online course content. This ensures that the student will have opportunities to acquire the knowledge, skills, perspectives, and practices needed to become an environmentally literate citizen. The online course should provide opportunities for each student to address environmental issues in their home, in their local community, or even at the global level.