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Course Overview

Mathematics, Grade 10, Applied

Identifying Information

Course Title: Foundations of Mathematics

Grade: 10

Course Type: Applied

Ministry Course Code: MFM2P

Credit Value: 1

Description/Rationale

This course enables students to consolidate their understanding of key mathematical concepts through hands-on activities and to extend their problem-solving experiences in a variety of applications. Students solve proportions and recognize when it is appropriate to use proportional reasoning and when it is not. Correct use of proportions is a very important skill in many learning, working, and leisure activities. Percentage, ratio, and rate work from earlier grades is captured in this new light, as are linear relationships. Similar triangles and trigonometric models for right-triangle problems provide new applications of proportions. Quadratic applications serve as contrasts to proportional reasoning. New algebraic skills for quadratics are introduced and practised. Algebraic skills are extended to include solution of linear systems and some quadratic equations, and some algebraic manipulation of quadratic expressions.

Unit Titles (Time + Sequence)

Unit 1	Proportional Reasoning	18 hours
Unit 2	Similar Triangles and Trigonometry	20.5 hours
Unit 3	Linear Systems	32 hours
Unit 4	Quadratic Functions	31 hours
Unit 5	Review and Summative Assessment:	8.5 hours

Unit Organization

Unit 1: Proportional Reasoning

Time: 18 hours

Description

Students practise the five steps in the inquiry/problem solving process: explore, hypothesize, model/formulate, manipulate/transform, infer/conclude, in a variety of contexts that can all be modelled by a proportional problem. Through exploration and generation of examples, students gain an increased depth of understanding of percent, ratio, and rate problems. Forming and testing hypotheses about the type, size, and units of results improves students’ intuition in situations involving proportions. Techniques for efficient creation of models for proportions are taught and practised. Manipulation of the equations resulting from substitution into a proportion model reinforces algebraic skills. At the infer/conclude stage, students are encouraged to communicate their findings, to reflect on the reasonableness of these findings, and to consciously reinforce or modify their hypotheses.

As with the study of any mathematical model, students should be exposed to appropriate and inappropriate applications of the model. Appropriate examples in this unit will include interest calculations, currency conversions, direct variation problems, and scale drawings. Inappropriate examples will include partial variation problems and distortion of figures resulting from irregular scales.

Overall Expectations: PRV.01.

Specific Expectations: PR1.01, PR1.02, PR1.03.

Unit 2: Similar Triangles and Trigonometry

Time: 20.5 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 are used to solve problems that result in right-angled triangles. The tangent ratio for the angle of inclination is connected to slope of a line, as students move from this unit to the next.

Overall Expectations: PRV.01, PRV.02, PRV.03.

Specific Expectations: PR1.01, PR2.01, PR2.02, PR2.03, PR3.01, PR3.02, PR3.03, PR3.04, PR3.05.

Unit 3: Linear Systems

Time: 32 hours

Description

This unit extends students' skills in solving linear equations to include fractional coefficients. They isolate a variable in a formula involving first degree terms. After this review of linear equations, students investigate the use of a linear system to model a comparison among several options. Points of intersection are found through numerical, graphical, and algebraic analysis and interpreted in the context of the problem. Formal algebraic methods of solving systems are motivated by the need for exactness and recognition of the challenge in reading exact values from a graph or a table. A balance among numerical, graphical, and algebraic analysis techniques develops conceptual understanding by appealing to various learning styles and skill strengths.

This unit also introduces the idea that relationships between two variables require a piecewise linear model when one constant rate changes to another. The implications of this, on tables of values, graphs, and equations, are investigated. A variety of contexts serve to illustrate the concepts and present opportunities to practise the skills.

Overall Expectations: LFV.01, LFV.02, LFV.03, PRV.03.

Specific Expectations: LF1.01, LF1.02, LF1.03; LF2.01, LF2.02, LF2.03, LF2.04, LF3.01, LF3.02, LF3.03, LF3.04, PR3.02, PR3.03.

Unit 4: Quadratic Functions

Time: 31 hours

Description

Students explore, hypothesize, model, manipulate, analyse, and make conclusions about data from quadratic situations, using primary and secondary sources. Given models are transformed and analysed. A rich contextual foundation is developed for subsequent algebraic studies.

Overall Expectations: QFV.01, QFV.02, QFV.03, PRV.02, LFV.01.

Specific Expectations: QF1.01, QF1.02, QF1.03, QF1.04, QF1.05, QF1.06, QF1.07, QF2.01, QF2.02, QF2.03, QF2.04, QF3.01, QF3.02, QF3.03, PR2.01, LF1.01, LF1.02, LF1.03.

Unit 5: Summative Assessment Activities

Time: 8.5 hours

Description

This unit is used to model a final assessment in Grade 10 Applied Mathematics. The activities have an entrepreneurial theme. Individual and group performance skills are assessed using traditional and performance based tasks, over a period of several days. Thirty percent of the final evaluation for the course will be based on this summative assessment unit.

Course Notes

This Course Profile demonstrates how to focus the Grade 10 Mathematics program on the key messages of the Grade 9 Profiles, re-visiting many of the fundamental ideas of mathematics:

· 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 application 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 help to ensure that students develop a solid set of knowledge and skills.

Other points to consider:

· When performing investigations, teachers are advised to 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 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 students must 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 full implementation.

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 and students with IEPs.

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. In forming an evaluative judgement, the teacher should consider students’ performances in the various Categories of the Achievement Chart separately from each other. The method of evaluation may vary in the various Categories of the Achievement Chart as suggested in the Gathering and Using Assessment Data chart on page 10 of this Profile.

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 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 these 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 in the policy document 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 paper and pencil questions are provided for each unit. Assessment tools are designed to allow students to demonstrate performance at the full range of learning (Levels 1 to 4).

It should be noted that:

· Tests that include only questions that 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 categories. 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 expectataions 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 Grade 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 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 still appropriate in the Knowledge and Application Categories.

Category	Stages in Learning to gather assessment data	Use of data for evaluation
Knowledge/ Understanding 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 actions (Explore, Hypothesize, Model/formulate, Manipulate, Infer/conclude – see Inquiry/Problem Solving chart for an explanation of the 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

Category

Stages in Learning to gather assessment data

Use of data for evaluation

Knowledge/
Understanding

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 actions (Explore, Hypothesize, Model/formulate, Manipulate, Infer/conclude – see Inquiry/Problem Solving chart for an explanation of the 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 to do	Specific examples of what to do
Explore	· Generate some examples. · Start to gather data. · Organize your work 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	· 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 should 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
http://www.edu.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, Foundations of Mathematics, MFM2P

Proportional Reasoning

Overall Expectations

PRV.01P

– solve problems derived from a variety of applications, using proportional reasoning;

PRV.02P

– solve problems involving similar triangles;

PRV.03P

– solve problems involving right triangles, using trigonometry.

Specific Expectations

Using Proportional Reasoning to Solve Problems from Applications

PR1.01P

– solve problems involving percent, ratio, rate, and proportion (e.g., in topics such as interest calculation, currency conversion, similar triangles, trigonometry, direct and partial variation related to linear functions) by a variety of methods and models (e.g., diagrams, concrete materials, fractions, tables, patterns, graphs, equations);

PR1.02P

– draw and interpret scale diagrams related to applications (e.g., technical drawings);

PR1.03P

– distinguish between consistent and inconsistent representations of proportionality in a variety of contexts (e.g., explain the distortion of figures resulting from irregular scales; identify misleading features in graphs; identify misleading conclusions based on invalid proportional reasoning).

Solving Problems Involving Similar Triangles

PR2.01P

– determine some properties of similar triangles (e.g., the correspondence and equality of angles, the ratio of corresponding sides) through investigation, using dynamic geometry software;

PR2.02P

– solve problems involving similar triangles in realistic situations (e.g., problems involving shadows, reflections, surveying);

PR2.03P

– 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

PR3.01P

– calculate the length of a side of a right triangle, using the Pythagorean theorem;

PR3.02P

– determine the measures of the sides and angles in right triangles, using the primary trigonometric ratios;

PR3.03P

– solve problems involving the measures of sides and angles in right triangles (e.g., in surveying, navigation);

PR3.04P

– determine the height of an inaccessible object in the environment around the school, using the trigonometry of right triangles;

PR3.05P

– describe applications of trigonometry in various occupations.

Linear Functions

Overall Expectations

LFV.01P

– apply the properties of piecewise linear functions as they occur in realistic situations;

LFV.02P

– solve and interpret systems of two linear equations as they occur in applications;

LFV.03P

– manipulate algebraic expressions as they relate to linear functions.

Specific Expectations

Applying Piecewise Linear Functions

LF1.01P

– explain the characteristics of situations involving piecewise linear functions (e.g., pay scale variations, gas consumption costs, water consumption costs, differentiated pricing, motion);

LF1.02P

– construct tables of values and sketch graphs to represent given descriptions of realistic situations involving piecewise linear functions, with and without the use of graphing calculators or graphing software;

LF1.03P

– answer questions about piecewise linear functions by interpolation and extrapolation, and by considering variations on given conditions.

Interpreting Systems of Linear Equations

LF2.01P

– determine the point of intersection of two linear relations arising from a realistic situation, using graphing calculators or graphing software;

LF2.02P

– interpret the point of intersection of two linear relations within the context of a realistic situation;

LF2.03P

– solve systems of two linear equations in two variables by the algebraic methods of substitution and elimination;

LF2.04P

– 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.

Manipulating Algebraic Expressions

LF3.01P

– write linear equations by generalizing from tables of values and by translating written descriptions;

LF3.02P

– rearrange equations from the form y = mx + b to the form Ax + By + C = 0, and vice versa;

LF3.03P

– solve first-degree equations in one variable, including those with fractional coefficients, using an algebraic method;

LF3.04P

– isolate a variable in formulas involving first-degree terms.

Quadratic Functions

Overall Expectations

QFV.01P

– manipulate algebraic expressions as they relate to quadratic functions;

QFV.02P

– determine, through investigation, the relationships between the graphs and the equations of quadratic functions;

QFV.03P

– solve problems by interpreting graphs of quadratic functions.

Specific Expectations

Manipulating Algebraic Expressions

QF1.01P

– multiply two binomials and square a binomial;

QF1.02P

– expand and simplify polynomial expressions involving the multiplying and squaring of binomials;

QF1.03P

– describe intervals on quadratic functions, using appropriate vocabulary (e.g., greater than, less than, between, from... to, less than 3 or greater than 7);

QF1.04P

– factor polynomials by determining a common factor;

QF1.05P

– factor trinomials of the form x² + bx + c;

QF1.06P

– factor the difference of squares;

QF1.07P

– solve quadratic equations by factoring.

Investigating the Connection Between the Graphs and the Equations of Quadratic Functions

QF2.01P

· – construct tables of values, sketch graphs, and write equations of the form y = ax² + b to represent quadratic functions derived from descriptions of realistic situations (e.g., vary the side length of a cube and observe the effect on the surface area of the cube);

QF2.02P

– 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;

QF2.03P

– explain the role of a, h, and k in the graph of y = a(x - h)² + k;

QF2.04P

– expand and simplify an equation of the form y = a(x - h)² + k to obtain the form y = ax² + bx + c.

Solving Problems Involving Quadratic Functions

QF3.01P

– obtain the graphs of quadratic functions whose equations are given in the form = a(x - h)² + k or the form y = ax² + bx + c, using graphing calculators or graphing software;

QF3.02P

– determine the zeros and the maximum or minimum value of a quadratic function from its graph, using graphing calculators or graphing software;

QF3.03P

– solve problems involving a given quadratic function by interpreting its graph (e.g., given a formula representing the height of a ball over elapsed time, graph the function, using a graphing calculator or graphing software, and 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?).

Acknowledgments

Identifying Information

Description/Rationale

Unit Titles (Time + Sequence)

Unit Organization

Unit 1: Proportional Reasoning

Unit 2: Similar Triangles and Trigonometry

Unit 3: Linear Systems

Unit 4: Quadratic Functions

Unit 5: Summative Assessment Activities

Course Notes

Teaching/Learning Strategies

Assessment and Evaluation

Gathering and Using Assessment Data

Accommodations

Accommodations for ESL/ESD students

Accommodations for students with learning disabilities

Resources

Professional Reading

Web Sites

Course Evaluation

Proportional Reasoning

Specific Expectations

Linear Functions

Specific Expectations

Quadratic Functions

Specific Expectations