Math, MCR3U, Grade 11, University Preparation
Functions This course introduces the mathematical concept of the function by extending students’ experiences with linear and quadratic relations. Students will investigate properties of discrete and continuous functions, including trigonometric and exponential functions; represent functions numerically, algebraically, and graphically; solve problems involving applications of functions; investigate inverse functions; and develop facility in determining equivalent algebraic expressions. Students will reason mathematically and communicate their thinking as they solve multi-step problems. Prerequisite: Principals of Mathematics, Grade 10 Academic
Summary of Units and Timelines
Below is the suggested sequence of course unit delivery as well as the recommended number of hours to complete the respective unit. For complete details of targeted expectations within each unit and activity, please see each Unit Overview found in the course profile.
|Unit Order||Unit Name||Suggested Time|
|Unit 0||Prerequisite Review of Concepts||5 hours|
|Unit 1||Introduction to Functions||15 hours|
|Unit 2||Algebraic Expressions||10 hours|
|Unit 3||Quadratic Functions||20 hours|
|Unit 4||Exponential Functions||17 hours|
|Unit 5||Functions & Applications of Trig||30 hours|
|Unit 6||Discrete Functions||18 hours|
|Total: 115 hours|
Mathematical Process Concepts Covered in this Course
Throughout this course, students will: Problem Solving
- Develop, select, apply, compare, and adapt a variety of problem-solving strategies as they pose and solve problems and conduct investigations, to help deepen their mathematical understanding;
Reasoning and Proving
- Develop and apply reasoning skills (e.g., use of inductive reasoning, deductive reasoning, and counter-examples; construction of proofs) to make mathematical conjectures, assess conjectures, and justify conclusions, and plan and construct organized mathematical arguments;
- Demonstrate that they are reflecting on and monitoring their thinking to help clarify their understanding as they complete an investigation or solve a problem (e.g., by assessing the effectiveness of strategies and processes used, by proposing alternative approaches, by judging the reasonableness of results, by verifying solutions);
Selecting Tools and Computational Strategies
- Select and use a variety of concrete, visual, and electronic learning tools and appropriate computational strategies to investigate mathematical ideas and to solve problems;
- Make connections among mathematical concepts and procedures, and relate mathematical ideas to situations or phenomena drawn from other contexts (e.g., other curriculum areas, daily life, current events, art and culture, sports);
- Create a variety of representations of mathematical ideas (e.g., numeric, geometric, algebraic, graphical, pictorial representations; onscreen dynamic representations), connect and compare them, and select and apply the appropriate representations to solve problems;
Communicating Communicate mathematical thinking orally, visually, and in writing, using precise mathematical vocabulary and a variety of appropriate representations, and observing mathematical conventions.
This course is designed to provide students with the necessary skills and knowledge as well as a development of an attitude that supports these skills and knowledge responsibly. The course design is geared to meet and exceed the prerequisite requirements for studying mathematics at the University level. The framework of course delivery focuses the “big pictures” or underlying principles, of mathematics as outlined on page 4 of the Ontario Ministry of Education document: The organization of the course is packaged into six distinct units which correspond with the Ontario Curriculum, Grades 11 and 12 Mathematics document. However, the number of units has been redesigned to what the author considers a more constructive development of material promoting a more manageable chucking of information from a students’ perspective. In addition, a pre-unit has been added to review and assess the knowledge and skills of students prior to commencing the actual grade 11 course material. Teachers will be providing an assessment for learning opportunity at the beginning of the course; after the pre-unit as recommended in the Growing Success 2010 document. Teachers will be able to use the results of this diagnostic to identify gaps in concepts for learners and provide opportunities to bridge these gaps preparing all learners to be successful in the Grade 11 Functions course by establishing a strong prerequisite foundation.
Teaching and Learning Strategies
Teachers will bring enthusiasm and varied teaching and assessment approaches to the classroom, addressing individual students’ needs and ensuring sound learning opportunities for every student. The activities offered should enable students to relate and apply these concepts to the social, environmental, and economical conditions and concerns of the world in which they live. Opportunities to relate knowledge and skills to these wider contexts will motivate students to learn in a meaningful way and to become life-long learners. Teachers will help students understand that problem solving of any kind often requires a considerable expenditure of time and energy and a good deal of perseverance. Teachers also will encourage students to investigate, to reason, to explore alternative solutions and to take the risks necessary to become successful problem solvers. Effective instructional approaches and learning activities draw on students’ prior knowledge, capture their interest, and encourage meaningful practise both inside and outside the classroom. Students will be engaged when they are able to see the connection between the mathematical concepts they are learning and their application in the world around them and in real-life situations. Due to its importance, students will have opportunities to learn in a variety of ways- individually, cooperatively, independently, with teacher direction, through hands-on experiences, and through examples followed by practice. The approaches and strategies teachers use will vary according to both the object of the learning and the needs of the students. Teachers will accomplish this in an on-line environment with the use of on-line simulations, animations, videos, discussion forums, live chat and other interactive objects.
Online & Offline Components
The design of this course is intended to offer a rich balance between online and offline elements. The following is a summary of the course components and their delivery format. Please refer to the individual unit outlines for specific details. Course content & instruction: online Communication between teacher and students: online & offline Collaboration between students: online Assessment & evaluation: online & offline Practise exercises, textbook work, readings etc: offline
Assessment & Evaluation for Student Achievement
As summarized in Growing Success 2010, the primary purpose of assessment and evaluation is to improve student learning. Information gathered through assessment helps teachers to determine students’ strengths and weaknesses in their achievement of the curriculum expectations in each course. This information also serves to guide teachers in adapting curriculum and instructional approaches to students’ needs and in assessing the overall effectiveness of programs and classroom practices. As part of assessment, teachers provide students with descriptive feedback that guides their efforts towards improvement. Evaluation refers to the process of judging the quality of student work on the basis of established criteria, and assigning a value to represent that quality. All curriculum expectations must be accounted for in instruction, but evaluation focuses on students’ achievement of the overall expectations. A students’ achievement of the overall expectations is evaluated on the basis of his or her achievement of related specific expectations. Teachers will use their professional judgement to determine which specific expectations should be used to evaluate achievement of overall expectations, and which ones will be covered in instruction and assessment but not necessarily evaluated. In order to ensure that assessment and evaluation are valid and reliable, and that they lead to the improvement of student learning, teachers must use assessment and evaluation strategies that:
- Address both what students learn and how well they learn;
- Are based both on the categories of knowledge and skills and on the achievement level descriptions given in the achievement chart
- Are varied in nature, administered over a period of time, and designed to provide opportunities for students to demonstrate the full range of their learning;
- Are appropriate for the learning activities used, the purposes of instruction, and the needs and experiences of the students;
- Are fair to all students;
- Accommodate students with special education needs, consistent with the strategies outlined in their Individual Education Plan;
- Accommodate the needs of students who are learning the language of instruction;
- Ensure that each student is given clear directions for improvement;
- Promote students’ ability to assess their own learning and to set specific goals
- Include the use of samples of students’ work that provide evidence of their achievement;
- Are communicated clearly to students and parents at the beginning of the school year and at other appropriate points throughout the school year.
The achievement chart for mathematics outlines four categories of knowledge and skills. They include; knowledge and understanding, thinking, communication and application. Teachers will ensure that student work is assessed and/or evaluated in a balanced manner with respect to the four categories, and that achievement of particular expectations is considered within the appropriate categories. A final grade is recorded for this course, and 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:
- Seventy percent of the grade will be based on evaluations conducted throughout the course. This portion of the grade should reflect the student’s most consistent level of achievement throughout the course, although special consideration should be given to more recent evidence of achievement. The seventy percent will be distributed in the following achievement chart categories: 20% Knowledge and understanding, 20% application, 15% communication and 15% thinking.
- Thirty percent of the grade will be based on a final evaluation in the form of an examination and administered towards the end of the course.
All students can succeed. Some students are able, with certain accommodations, to participate in the regular course curriculum and to demonstrate learning independently. Accommodations allow access to the course without any changes to the knowledge and skills the student is expected to demonstrate. The accommodations required to facilitate the student’s learning can be identified by the teacher, but recommendations from a School Board generated Individual Education Plan (IEP) if available can also be consulted. Instruction based on principles of universal design and differentiated instruction focuses on the provision of accommodations to meet the diverse needs of learners. Examples of accommodations (but not limited to) include:
- Adjustment and or extension of time required to complete assignments or summative tasks
- Providing alternative assignments or summative tasks
- Use of scribes and/or other assistive technologies
- Simplifying the language of instruction
Teachers will bring additional resources and teaching materials that provide a rich and diverse learning environment. Units in this course profile make specific reference to the intended textbook for this course but can be substituted for any relevant and approved text. Speijer, Jacob. McGraw-Hill Ryerson Functions 11. Toronto: McGraw-Hill Ryerson, 2009. Small, Marian. Nelson Functions 11. Toronto: Nelson Education, 2008. Policy Document: The Ontario Curriculum, Grades 11 to 12, Mathematics, 2007 Ministry of Education of Ontario Reference: http://www.edu.gov.on.ca/eng/curriculum/secondary/math1112currb.pdf