Course Overview
Principles of Mathematics, Grade 10, Academic
Identifying Information
Course Title: Principles of Mathematics
Grade: 10
Course Type: Academic
Ministry Course Code: MPM2D
Credit Value: 1
Description/Rationale
This course enables students to broaden their understanding of relations, extend their skills in multi-step problem solving, and continue to develop their abilities in abstract reasoning. Students will model linear and quadratic relationships arising from a variety of contexts. Using trigonometric ratios and analytic geometry techniques, students will learn how to find exact measures in geometric contexts, as opposed to the approximate measures they have found using scale drawings and measurement tools. Geometric relationships investigated in Grade 9 will be confirmed, analytically, in specific cases, and students will be introduced to proof in general. Algebraic skills will be extended to generate factored, expanded, and completed square forms of quadratic expressions, and to solve linear systems and quadratic equations. Fundamental mathematical ideas of modelling, patterning, optimization, and superimposing a grid onto a geometric situation are reinforced. Connections among the various strands of the course are intentionally developed.
Unit Titles (Time + Sequence)
|
Part A: Assessment |
Preparation and Gearing Up |
5 hours |
|
Unit 1 |
Similar Triangles and Trigonometry |
26 hours |
|
Unit 2 |
Analytic Geometry |
21 hours |
|
Unit 3 |
Linear Systems |
21 hours |
|
Part B: Assessment |
Review and Summative Assessment Tasks for Units 1, 2, and 3 (Bermuda Triangle) |
5 hours |
|
Unit 4 |
Quadratic Functions |
27 hours |
|
Part C: Assessment |
Review and Summative Assessment (Unit 4 and Selected Other Expectations) |
5 hours |
Unit Organization
Part A Assessment: Preparation and Gearing Up
Time: 5 hours
Description
Students start the course with four activities that prepare them for effective performances in the four Categories of the Achievement Chart, one per day in a semestered school. These activities help students to gear up for the demands of the Grade 10 program by reviewing previous learnings and connecting them in new ways. The Application activity requires students to connect pieces of knowledge and skills from Grade 9. The Communication activity foreshadows geometric scenarios needed later in the course and emphasize the need for teamwork, self-confidence, and perseverance. The Knowledge/Understanding activity provides diagnostic information for the teacher and emphasizes the importance of independence. The Inquiry/Problem solving activity reviews the necessary actions: explore, hypothesize, model, transform, and conclude, and emphasize the importance of perseverance. Through these activities, the stage is set for a balance of appropriate assessment tasks throughout the course. The Learning Skills necessary for success are reinforced as teachers acquaint students with their expectations in these areas.
Unit 1: Similar Triangles and Trigonometry
Time: 26 hours
Description
Students are introduced to applications of similar triangles and trigonometry through a variety of activities that use concrete materials and allow students to move about inside and outside the classroom. Primary trigonometric ratios, Sine and Cosine Laws are used to solve problems that are modelled by right-angled or acute triangles. As students move from the first unit to the second unit, they investigate how the tangent ratio for the angle of inclination is connected to slope of a line.
Overall Expectations: TRV.01, TRV.02, TRV.03.
Specific Expectations: TR1.01, TR1.02, TR1.03, TR1.04, TR2.01, TR2.02, TR2.03, TR3.01, TR3.02, TR3.03, TR3.04, TR3.05.
Unit 2: Analytic Geometry
Time: 21 hours
Description
This unit provides contexts for developing formulas for midpoint, distance between points, and circles centred at the origin. Then geometric relationships investigated in Grade 9 mathematics are confirmed through the use of the Cartesian system and formulas. Properties of triangles and quadrilaterals are investigated analytically.
Overall Expectations: AGV.02, AGV.03.
Specific Expectations: AG2.01, AG2.02, AG2.03, AG2.04, AG3.01, AG3.02, AG3.03.
Unit 3: Linear Systems
Time: 21 hours
Description
This unit will focus on the use of two linear equations to model a problem. In some cases, both lines are graphical models where the point of intersection of the lines has meaning in the context of the problem. Points of intersection will be found through numerical, graphical, and algebraic analysis. In other cases, only parts of two lines are needed to model a single situation. These result in consideration of a range of values for solution to an optimization problem through linear programming analysis. This unit also contains multi-step problems in analytic geometry which require solution of a linear system.
Overall Expectations: AGV.01, AGV.02.
Specific Expectations: AG1.01, AG1.02, AG1.03, AG2.03, AG2.04.
Part B Assessment: Review and Summative Assessment Tasks for Units 1, 2, and 3 (Bermuda Triangle)
Time: 5 hours
Description
A series of rich problems, with a Bermuda Triangle theme, are posed in this section. Solution of these problems requires use of learning from the first three units of this Profile. All four Categories of the Achievement Chart will be able to be assessed using these problems. By placing these summative assessment activities before the last Unit, the teacher will have time in the remainder of the course to remediate critical skills that are found to be weak before the end of the course.
Unit 4: Quadratic Functions
Time: 27 hours
Description
This unit enables students to broaden their understanding of relations, extend their skills in multi-step problem solving, and continue to develop their abilities in abstract reasoning. Students will gather, organize, manipulate, and analyse data from primary and secondary sources to model and communicate results about quadratic situations. A variety of problems will be studied to ensure that students will gain depth of understanding of quadratics through meeting the same specific expectations in different contexts. Students will conduct investigations to verify or refute their own conjectures about relationships, using lines or curves of best fit, tables, and pattern descriptions. They will communicate their findings and describe trends. A rich foundation for quadratics, built on experiences from a variety of real world contexts, will be built before subsequent algebraic studies are undertaken.
Overall Expectations: QFV.01, QFV.02, QFV.03, QFV.04.
Specific Expectations: QF1.01, QF1.02, QF1.03, QF1.04, QF1.05, QF2.01, QF2.02, QF2.03, QF2.04, QF3.01, QF3.02, QF3.03, QF3.04, QF4.01, QF4.02, QF4.03, QF4.04.
Part C Assessment: Review and Summative Assessment (Unit 4 and Selected Other Expectations)
Time: 5 hours
Description
Summative assessment performance tasks for Unit 4 should definitely be included at this time. There is also time for a formal exam on part or all of the course. This Profile does not contain sample exam questions since any of the problems posed throughout the Profile could be used here if they were not used earlier. It would also be possible to use some of the Bermuda Triangle performance tasks at the end of the course, if they were not used earlier.
Course Notes
This course Profile demonstrates how to focus the Grade 10 Mathematics program on the key messages of the Grade 9 Profiles.
· The high school mathematics curriculum is designed to help students understand the power of mathematics in modelling authentic problems and situations, and acquire the important skills needed to create, interpret, and analyse such models.
· The learning of skills is enhanced for students when the need for the skill arises from a contextual setting. Teachable moments can be captured in the responses to the ideas and questions posed by students. Once the need for a skill has been identified, it is important that the skill be developed and practised.
· The use of a range of strategies is expected. Teacher-directed strategies such as Socratic lessons or examples followed by practice are effective in the teaching of skills. Other strategies are effective in implementing the inquiry-related expectations of the curriculum. These may include investigations and opportunities for students to communicate their processes, reasoning, and findings.
Exploration allows students to develop an understanding of the applications of mathematical principles. However, it is critically important that students are guided to enunciate and understand the essential mathematical principles themselves. The teacher’s role is important in facilitating this process of bringing closure to an activity (extracting the mathematics from the context). Carefully providing opportunities for students’ communication of their developing understanding, along with frequent formative feedback, will ensure that students develop a solid set of knowledge and skills.
Other points to consider:
·
Students who took the Applied course
in Grade 9 may, initially, have less experience and confidence with some forms
for formulas and equations, than students who took the Academic course. Care
should be taken to go back to the basic formula or concept, before using more
abstract forms. For example, start with 
before using 
or 
.
· When performing investigations, it is important that teachers carry out the experiments themselves, beforehand. Considerations should include timing, questions to pose, means of drawing closure, prerequisite skills, appropriate connections and links to other disciplines, and the choosing of investigations having regional or current interest.
· Technology can be useful in learning, doing, and assessing student achievement in mathematics. When using technology for an activity, teachers are advised to practise its use beforehand. This profile will identify which graphic calculator and dynamic geometry software skills are needed for each activity. Occasionally, teachers may wish to demonstrate the use of technology as a tool for gathering, organizing and displaying data, and at other times it is appropriate that students have their own hands on the technology. Technology is used in the Course Profile to support concept development and to extend applications. Technology is not used to replace skill development.
· Use of the Achievement Levels Chart of Mathematics is the basis of assessment and reporting of all aspects of the course.
· The implementation of Grade 10 Mathematics is a process, not an event. Through consistent effort, gradual progress will be made towards the full implementation of the curriculum.
· It would be possible to sequence the Similar Triangles and Trigonometry Unit second, after Analytic Geometry, if use of the computer lab cannot be arranged near the beginning of the course. Note that the Applied profile sequences the Trigonometry Unit second, avoiding conflict with the Academic profile for computer lab space, if it is limited.
Teaching/Learning Strategies
Only through the use of a wide variety of teaching, learning, and assessment strategies and tools can the wide range of expectations in this course be addressed.
Teachers will:
· include a balance of whole class, small group and individual instruction;
· include a balance of student-centred and teacher-directed activities;
· provide students with materials, technological tools and software for use in experiments, demonstrations, and investigations;
· address a variety of learning styles in each unit;
· plan so that time is spent engaging students in the solution of contextual problems;
· be accountable for addressing the overall and specific expectations in their planning, and accountable for tracking student progress in the expectations;
· assume a variety of roles in the classroom, including both director of learning and guide or facilitator of learning;
· provide many opportunities for students to demonstrate their learning of the course expectations;
· ensure that the culmination of an activity helps the students to build a solid understanding of the mathematical concepts arising from that activity and sets the stage for future learning;
· prompt at the beginning of an activity, provide suggestions in the middle, and support and challenge at the end, as needed by individual students and by the class as a whole;
· provide verbal instruction to accompany written procedures to avoid the frustration and uncertainty that may otherwise undermine the learning opportunities afforded by a complex task;
· use learning/performance tasks that are designed to link several expectations and give the students occasion to demonstrate their optimal levels of achievement through the demonstration of skill acquisition, the communication of results, the ability to pose extending questions following an inquiry, and to provide the solution to unfamiliar problems;
· provide remediation or extension opportunities;
· provide opportunities for students to practise or extend their skills and knowledge, outside of the classroom;
· provide regular, informal assessment which provides the feedback that students need in order to improve their achievement;
· modify instructional and assessment strategies for special needs students.
Students
will:
· develop increasing responsibility for their own learning;
· follow examples and Socratic lesson developments and take notes provided by the teacher ;
· carry out investigations and engage in the inquiry process;
· demonstrate an understanding of concepts, and ability to select and perform algorithms accurately in order to solve problems;
· practise prerequisite skills;
· explore, hypothesize, formulate, manipulate, infer/conclude, and communicate during an inquiry;
· engage in explorations involving the use of technology (e.g., graphing software, dynamic geometric software, databases, the Internet, statistical programs, spreadsheets, and multimedia resources) and the collection of data;
· apply individual and group learning skills;
· pose and answer questions in a context;
· describe the patterns that emerge verbally, algebraically, and visually (using tables, graphs, and posters).
Assessment and Evaluation
Assessment is a systematic process of collecting information or evidence about student learning; evaluation is the judgement we make about the assessments of student learning based on established criteria. This profile will focus on providing specific examples of assessment strategies and tools and general statements about how these assessments might be used in evaluation. Evaluation requires that the teacher not simply average marks. The method of evaluation may vary in the various Categories of the Achievement Chart as suggested in the Gathering and Using Assessment Data chart.
The focus of this course is on inquiry, problem solving, communication, acquisition of high levels of knowledge and skills, and application of mathematics. Knowledge and understanding continue to be important. Assessment looks at students meeting course expectations at a variety of levels, with an emphasis on growth over time. Assessment should be used to gather information for diagnostic, formative and summative purposes. It is important to note that assessment and evaluation will be criterion referenced, comparing student performance to the Ministry standard, not to other students. Level 3 is defined as the provincial standard. A student achieving at this level is well-prepared for work in the Grade 11 U or 11 U/C course. Level 4 performance requires a consistent, but not constant, pattern of well-communicated higher level thinking and not simply technically correct solutions. Level 4 does not require a student to perform beyond grade level expectation.
Assessment strategies and tools must address the variety of teaching and learning styles as well as the variety of expectations. High quality assessment can measure individual and group performance, and individual performance within a group. A balanced assessment program will include various methods:
· journals, portfolios, and projects
· performance assessments and presentations
· conferencing
· tests and quizzes
Assessment tools to be used throughout the course
include:
· the four level Achievement Chart
· rubrics (both teacher-created and student-generated)
· checklists
· anecdotal comments
· objective marking schemes
When teachers use a variety of these assessment tools, it is necessary to ensure that a consistent standard is maintained. That is, a 70-79% performance using an objective marking scheme should be equivalent to a Level 3 performance. Teachers may find it more appropriate to use rubrics to assess Inquiry/Problem Solving and Communication, and objective scales for Knowledge/Understanding and Application, as they are beginning to gather data in the Categories of the Achievement Chart. In doing so, it is important that they keep in mind that Level 3 and 70-79% are the provincial standard. Performance tasks and tests should be set with the Expectations as the criteria for this standard.
A selection of assessment tools has been designed or identified to accompany specific assessment activities. Teachers are encouraged to use them, then develop similar tools for other assessment activities. Some suggestions for increasing scoring consistency include:
· involve other teachers in the department in the creation of rubrics for assessment
· involve students in the setting of criteria, and the use of self- and peer assessments
· gather exemplars of student work at the four levels, so that teachers and students can get a better image of what achievement at these levels looks like
Assessment of the expectations, using the four Levels of the Achievement Chart, is ongoing throughout the course Profile. A summative performance activity and summative pencil and paper questions are provided for each unit. Assessment tools are designed to allow students to demonstrate performance at the full range of their learning (Levels 1 to 4).
It should be noted that:
· Tests including only questions which ask students to perform algorithms and apply their knowledge do not necessarily offer an opportunity for students to demonstrate Level 4 performance.
· It is often easier to pose questions with the expectation of Level 1 to 4 responses in the Inquiry/Problem Solving and Communication Categories of the Achievement Chart than the Knowledge/Understanding and routine Application Categories.
· Teachers must expand their understanding of Applications to include non-routine applications. This newer view of Applications requires a shift from thinking of them as being tied to specific content, to applications of mathematics, in general.
· The issue of communication is complex. Teachers need to ask students to communicate their understanding of their knowledge, their stages of thought in an inquiry, and their process of applying mathematics to a problem, in order to assess Level 1 to 4 performances in the other three Categories of the Achievement Chart. Then, they need to report on the Communications Category separately from those. See the chart below for details concerning the various aspects of communication.
· New tools and strategies are needed to offer students opportunities to be assessed in the Inquiry, Communication, or Application (the process) Categories of the Achievement Chart.
· The expectations of the course include a wide range of skills, all of which must be addressed. This Profile has labelled some skills as critical, with the belief that students should be encouraged to practise those skills, on their own time, persevering until those skills have been mastered. To ensure that learning of these critical skills has happened, teachers will have to keep track of which students have and have not demonstrated the required learning. Those who did not demonstrate the expected level of achievement earlier should be re-tested after more learning has happened. Students might be given repeated opportunities to demonstrate acquisition and retention of all critical skills of the course if a “Part A” on each written test includes one question drawn from previous units.
Many suggestions for journal entries have been offered throughout the Activities. Teachers may wish to have students set up a section in their regular notebook for journal entries or use a separate notebook. The journals could be used for explaining concepts, summarizing the learning from an inquiry, recording brainstorming or hypotheses, reflecting, etc. Teachers will sometimes ask students to respond to specific questions or record stages of a structured investigation in their journals. Other times, the teacher will provide students with opportunities to respond to open-ended questions.
Journal entries can be assessed both formally and informally. Criteria within the Categories of the Achievement Chart are suggested for times when the teacher may want to assess journal work formally. For informal assessment the teacher may wish to:
· Have several students share their entries orally while other students add new ideas or missing information to their own entries.
· Collect the journals and offer written comments, but no mark, on the student's entry.
· Have students trade their journals with a partner and allow some time for discussion between the pair about the completeness and clarity of their entries.
The teacher should always let students know who will be reading specific journal entries before they start their entries.
This profile contains more assessment suggestions than it would be reasonable for any teacher to use in one course. The expectation is that teachers will try a variety of strategies and tools so that an informed decision can be made about which of these works best, for them, in the various Categories of the Achievement Chart. Some of the assessment suggestions in this profile will appeal to teachers at each of the three stages of implementation, as outlined in the chart below.
Assessment activities in this profile will be based on the following analysis of how Mathematics teachers could gather and use assessment data for the Categories in the Achievement Chart.
Gathering
and Using Assessment Data
The Grades 9 and 10 course expectations will be assessed using a wide range of assessment techniques. The Categories of the Achievement Chart can be used as organizers for the assessments, and the recording of assessment data. Teachers will be at different stages of readiness for full implementation of the range of assessment strategies suggested in this Profile. In the following chart, an example has been suggested for each category describing three stages of implementation.
Suggestions for evaluation of assessment data in the Categories include “weighted mean averages” and “medians of most recent performances”, depending on the Category. Recent medians are suggested in the Inquiry and Communication Categories, where students can be expected to improve their performances as the course proceeds and they learn how to effectively employ their new knowledge, skills, and vocabulary. The practice of using mean averages is appropriate in the Knowledge and Application Categories.
|
Category |
Stages
in Learning to gather assessment data |
Use
of data for evaluation |
|
Knowledge How well does a student understand a concept? How consistent is a student in selecting and performing appropriate algorithms? |
Stage 1: Gather marks from a test and convert to Levels using descriptors Stage 2: Score a test using a rubric Stage 3: Provide informal and formal feedback on Knowledge and Understanding demonstrated through a wide variety of types of activities |
Weighted mean Later stages – recent median |
|
Inquiry/Problem Solving How well can a student go about a mathematical inquiry? |
Stage 1: Use activities and rubrics in the profiles for instruction and assessment Stage 2: Incorporate the necessary inquiry actions (Explore, Hypothesize, Model/formulate, Manipulate, Infer/conclude – see Inquiry/Problem Solving chart for an explanation of these actions) into Stage 1 activities, and others like them Stage 3: Gather or create your own inquiries, addressing course Expectations |
Recent median: [Informal feedback can be used for formative assessment and formal feedback can be recorded] |
|
Communication How well can a student communicate mathematical thought? |
Stage 1: Incorporate communications questions into written tests and journals and provide informal feedback on a variety of communication modes (presentations, written submissions, posters, etc.) Stage 2: Through the use of the Activities in the Profiles, assess communication using the appropriate parts of the rubrics in the profiles. Stage 3: Use a wide variety of communication modes for both informal and formal assessment using rubrics based on: · Ability to read and interpret mathematics · Ability to integrate narrative and mathematical forms of communication · Quality of reporting on processes used · Degree of clarity in explanations and justifications in reporting on problem solving · Appropriateness of use of mathematical vocabulary · Correctness of use of mathematical symbols, labels, and conventions |
Recent median [Informal feedback can be used for formative assessment and formal feedback can be recorded] |
|
Application Is a student able to apply his/her knowledge to routine and non-routine situations? How well can a student connect these bits of knowledge together? |
Stage 1: Marks on routine application questions on a test are converted to Levels using descriptors. Teachers offer informal feedback on solution of non-routine problems. Stage 2: Score test questions using a rubric for applications found within the activities. Stage 3: Provide informal and formal feedback on applications demonstrated through a wide variety of types of activities, encouraging solution of non-routine problems |
Stage 1: Weighted mean Later stages – recent median |
To help both teachers and students focus on the component actions that are needed for successful inquiries and problem solving, the following chart is partially completed. More entries can be made in the third column as students learn more within a course and as they move from course to course. A generic rubric, showing the four Levels of Achievement against the actions outlined below, could be created and shared with students.
|
Inquiry/Problem Solving |
||
|
Action |
What you need to do |
Specific examples of what to do |
|
Explore |
· Try things. · Generate some examples. · Start to gather data. · Your work may not be well organized with a sequential flow. |
In a geometric context: - make a drawing and make measurements or deductions - try to use a known formula - try to develop a formula In a non-geometric context: - look for a pattern - extend patterns |
|
Hypothesize |
Make an educated guess, based on what you can deduce using outside knowledge and on your exploration. |
Get yourself to the stage where you can say, “I think that it’s….so if I create …. type of a model, I will be able to confirm or deny my hypothesis…” |
|
Model/ Formulate |
Form a mathematical model. |
Create a table of values, a scatterplot, an equation, an algorithm… |
|
Manipulate/ Transform |
Work with your model to get the information you need. |
If you have a table of values or a scatterplot: - interpolate or extrapolate - refine your scale - zoom in If you have an equation: - factor - complete the square - solve If you have an algorithm: - carry out a variety of examples |
|
Infer/ Conclude |
Decide what inferences and conclusions you can make using your model and logic. Decide if your model needs adjustment Communicate your findings. |
- Look back to your hypothesis to see if you have denied it or confirmed it partially or fully. - Decide if you need to gather more data or refine your model. - See if your findings make sense when you consider all that you know about the situation. - Articulate your findings precisely. |
Generic rubrics for Thinking, Inquiry, and Problem Solving are being developed. These describe Levels of Achievement for criteria which describe the actions named above. It is suggested that a generic rubric be given to students for reference throughout the course. Consistent references to the same criteria and descriptors of Levels will help learning and performance.
When students are given TIPS performance tasks, the teacher should identify which criteria from the generic rubric will be assessed. To mark student work, teachers will develop a rubric specific to the task, using the criteria identified for the students. The Level descriptors will describe student work on this task. To share with students the detailed rubric for a specific performance task before the task is completed is not appropriate since the information in the rubric would essentially tell students things they should have to think of themselves. Instead, teachers should use the detailed rubric as their marking scheme.
It is expected that teachers will record their assessment data for Expectations in Levels or percentages, or a combination of the two. It is important that whichever system is used, a consistent understanding of the type of performance that is intended by each mark be uppermost in the minds of teachers, students and parents. At reporting time, it will be necessary to convert the assessment data to a percentage grade. Learning skills will be reported separately from the mark based on demonstration of Expectations.
Accommodations
The following accommodations can be made throughout the course.
Accommodations for ESL/ESD students
· Have students work with partners, peer tutors, or classmates who share the same linguistic background.
· Provide extensive student/teacher conferencing.
· Use peer conferencing to reinforce instructions/information.
· Ask an ESL/ESD teacher to review questions, assignments, or assessment instruments.
· Provide sets of reference notes, outlines of critical information, and models of charts, timelines, or diagrams.
· Reinforce main ideas by using think/pair/share.
· Pair written instruction with verbal instructions.
· Use key visuals to illustrate definitions for the students’ dictionary of key words.
· Simplify instructions.
· Highlight key words or phrases.
· Brainstorm in groups in first language if English is limited.
· Provide opportunities for students to practise oral presentation skills.
· Provide visual/auditory cues.
Accommodations for students with learning disabilities
· Provide extensive student/teacher conferencing.
· Pair students.
· Provide a list of terminology (possibly simplified) before an activity begins.
· Modify handouts in terms of language and content used, and in terms of size and easy-to-read font.
· Allow assignments to be completed in alternate formats or in longer timelines.
· Keep manipulatives, grid paper, formulas sheets, and other aids available for needs that arise.
· Contact parent/guardian for support and suggestions.
· Provide oral preplanning of activities with students.
Resources
Professional Reading
Stenmark, Jean Kerr, ed. Mathematics Assessment: Myths, Models, Good Questions and Practical Suggestions. NCTM, 1991, ISBN 0-87353-339-9
Stenmark, Jean Kerr. Assessment
Alternatives in Mathematics: An Overview of Assessment Techniques That Promote
Learning. Assessment Committee of the California Mathematics Council.
ISBN 0912511540
Web Sites
Getting
Assessment Right: Mathematics, Data Based Directions
http://www.dbdirections.com
Source page for the Harvard
Balanced Assessment Project
gseweb.harvard.edu/~etc/ba/index.html#about
More information about balanced
assessment
www.educ.msu.edu/MARS/services/what.html
Course Evaluation
Course improvement should be viewed as an ongoing and collaborative process among mathematics teachers. As new resources, new technology, and new insights on the programs develop, teachers will adapt their programs to better serve the needs of their students.
To meet these goals, teachers should evaluate the effectiveness of their courses using a variety of information sources. While students’ performances on summative tasks are obvious indicators of success, many other sources exist. These include students’ reflections on their learning in their mathematics journals, parental feedback, and performance of students in subsequent mathematics courses, as well as other subject disciplines which build on Grade 10 Mathematics.
Seventy per cent of the grade will be based on assessments and evaluations conducted throughout the course. Thirty per cent of the grade will be based on a final evaluation in the form of an examination, performance, essay, and/or other method of evaluation suitable to the course content and administered towards the end of the course.
Anecdotal
evidence can be gathered from observing the following indicators:
· the care students take in their work;
· students’ efforts to complete their work and seek help as needed;
· students’ pursuit of extension activities;
· students’ growth in independence and persistence when completing tasks.
Coded Expectations, Principles of Mathematics, MPM2D
Quadratic Functions
Overall Expectations
QFV.01D
– solve quadratic equations;
QFV.02D
– determine, through investigation, the relationships between the graphs and the equations of quadratic functions;
QFV.03D
– determine, through investigation, the basic properties of quadratic functions;
QFV.04D
– solve problems involving quadratic functions.
Specific Expectations
Solving Quadratic Equations
QF1.01D
– expand and simplify second-degree polynomial expressions;
QF1.02D
– factor polynomial expressions involving common factors, differences of squares, and trinomials;
QF1.03D
– solve quadratic equations by factoring and by using graphing calculators or graphing software;
QF1.04D
– solve quadratic equations, using the quadratic formula;
QF1.05D
– interpret real and non-real roots of quadratic equations geometrically as the x-intercepts of the graph of a quadratic function.
Investigating the Connections Between the Graphs and the Equations of Quadratic Functions
QF2.01D
– identify the effect of simple transformations (i.e., translations, reflections, vertical stretch factors) on the graph and the equation of y = x2, using graphing calculators or graphing software;
QF2.02D
– explain the role of a, h, and k in the graph of y = a(x - h)2 + k;
QF2.03D
– express the equation of a quadratic function in the form y = a(x - h)2 + k, given it in the form y = ax2 + bx + c, using the algebraic method of completing the square in situations involving no fractions;
QF2.04D
– sketch, by hand, the graph of a quadratic function whose equation is given in the form y = ax2 + bx + c, using a suitable method [e.g., complete the square; locate the x-intercepts if the equation is factorable; express in the form y = ax(x - s) + t to locate two points and deduce the vertex].
Investigating the Basic Properties of Quadratic Functions
QF3.01D
– collect data that may be represented by quadratic functions, from secondary sources (e.g., the Internet, Statistics Canada), or from experiments, using appropriate equipment and technology (e.g., scientific probes, graphing calculators);
QF3.02D
– fit the equation of a quadratic function to a scatter plot, using an informal process (e.g., a process of trial and error on a graphing calculator), and compare the results with the equation of a curve of best fit produced by using graphing calculators or graphing software;
QF3.03D
– describe the nature of change in a quadratic function, using finite differences in tables of values, and compare the nature of change in a quadratic function with the nature of change in a linear function;
QF3.04D
– report the findings of an experiment in a clear and concise manner, using appropriate mathematical forms (e.g., written explanations, tables, graphs, formulas, calculations), and justify the conclusions reached.
Solving Problems Involving Quadratic Functions
QF4.01D
– determine the zeros and the maximum or minimum value of a quadratic function, using algebraic techniques;
QF4.02D
– determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software;
QF4.03D
– solve problems related to an application, given the graph or the formula of a quadratic function (e.g., given a quadratic function representing the height of a ball over elapsed time, answer questions such as the following: What is the maximum height of the ball? After what length of time will the ball touch the ground? Over what interval is the height of the ball greater than 3 m?).
Analytic Geometry
Overall Expectations
AGV.01D
– model and solve problems involving the intersection of two straight lines;
AGV.02D
– solve problems involving the analytic geometry concepts of line segments;
AGV.03D
– verify geometric properties of triangles and quadrilaterals, using analytic geometry.
Specific Expectations
Using Linear Systems to Solve Problems
AG1.01D
– determine the point of intersection of two linear relations graphically, with and without the use of graphing calculators or graphing software, and interpret the intersection point in the context of a realistic situation;
AG1.02D
– solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination;
AG1.03D
– solve problems represented by linear systems of two equations in two variables arising from realistic situations, by using an algebraic method and by interpreting graphs.
Solving Problems Involving the Properties of Line Segments
AG2.01D
– determine formulas for the midpoint and the length of a line segment and use these formulas to solve problems;
AG2.02D
– determine the equation for a circle having centre (0, 0) and radius r, by applying the formula for the length of a line segment; identify the radius of a circle of centre (0, 0), given its equation; and write the equation, given the radius;
AG2.03D
– solve multi-step problems, using the concepts of the slope, the length, and the midpoint of line segments (e.g., determine the equation of the right bisector of a line segment, the coordinates of whose end point are given; determine the distance from a given point to a line whose equation is given; show that the centre of a given circle lies on the right bisector of a given chord);
AG2.04D
– communicate the solutions to multi-step problems in good mathematical form, giving clear reasons for the steps taken to reach the solutions.
Using Analytic Geometry to Verify Geometric Properties
AG3.01D
– determine characteristics of a triangle whose vertex coordinates are given (e.g., the perimeter; the classification by side length; the equations of medians, altitudes, and right bisectors; the location of the circumcentre and the centroid);
AG3.02D
– determine characteristics of a quadrilateral whose vertex coordinates are given (e.g., the perimeter; the classification by side length; the properties of the diagonals; the classification of a quadrilateral as a square, a rectangle, or a parallelogram);
AG3.03D
– verify geometric properties of a triangle or quadrilateral whose vertex coordinates are given (e.g., the line joining the midpoints of two sides of a triangle is parallel to the third side; the diagonals of a rectangle bisect each other).
Trigonometry
Overall Expectations
TRV.01D
– develop the primary trigonometric ratios, using the properties of similar triangles;
TRV.02D
– solve trigonometric problems involving right triangles;
TRV.03D
– solve trigonometric problems involving acute triangles.
Specific Expectations
Developing the Primary Trigonometric Ratios
TR1.01D
– determine the properties of similar triangles (e.g., the correspondence and equality of angles, the ratio of corresponding sides, the ratio of areas) through investigation, using dynamic geometry software;
TR1.02D
– describe and compare the concepts of similarity and congruence;
TR1.03D
– solve problems involving similar triangles in realistic situations (e.g., problems involving shadows, reflections, surveying);
TR1.04D
– define the formulas for the sine, the cosine, and the tangent of angles, using the ratios of sides in right triangles.
Solving Problems Involving the Trigonometry of Right Triangles
TR2.01D
– determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios;
TR2.02D
– solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation);
TR2.03D
– determine the height of an inaccessible object in the environment around the school, using the trigonometry of right triangles.
Solving Problems Involving the Trigonometry of Acute Triangles
TR3.01D
– determine, through investigation, the relationships between the angles and sides in acute triangles (e.g., the largest angle is opposite the longest side; the ratio of side lengths is equal to the ratio of the sines of the opposite angles), using dynamic geometry software;
TR3.02D
– calculate the measures of sides and angles in acute triangles, using the sine law and cosine law;
TR3.03D
– describe the conditions under which the sine law or the cosine law should be used in a problem;
TR3.04D
– solve problems involving the measures of sides and angles in acute triangles;
TR3.05D
– describe the application of trigonometry in science or industry.
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