Senior Mathematics Curriculum Revision 4 Supporting students and teachers by keeping Ontario’s K - 12 curriculum current and relevant College Math Project

Senior Mathematics Curriculum Revision

4

Supporting students and teachers by keeping Ontario’s K - 12 curriculum current and relevant

College Math Project ForumJune 15, 2006

Anthony AzzopardiCurriculum and Assessment Policy BranchMinistry of Education

What is Curriculum Review?

A staged process to review Kindergarten to Grade 12 curriculum documents by discipline area that:• builds on the quality curriculum currently in place• ensures that the curriculum remains current and relevant

Curriculum Review Process

• integrate review of elementary and secondary curriculum policy documents

• have parallel revision processes for English and French language curriculum

• involve teachers, principals, board staff, subject experts, education stakeholders, parents, students and sector representatives.


* Mandatory Implementation

Revision and Feedback Consultation

Analysis and Synthesis

Editing, Publication and Distribution

Sept.2003

Sept.2004

Sept.2005

Sept.2006

Sept.2007

**

Grade 11

Grades 1 - 10

Grade 12 *

Subject /Division

Associations

Focus Groups

Other Consultations

and Input

Analysis / Synthesis

Feedback Consultations


Technical Analysis

Revision / Feedback

Achievement Charts Research

Grade 9Academic

Grade 10Academic

Grade 11 WMath for

Everyday Life

Grade 10Applied

Grade 9Applied

Grade 11 U/CFunctions

Grade 11 CPersonalFinance

Grade 12CCollege and

Apprenticeship

Grade 12CMath for College

Technology

Grade 12UData

Management

Grade 12UAdvanced Functions

Grade 11 UFunctions and

Relations

Grade 12UGeometry and

Discrete

Grade 12 WMath for

Everyday Life

Ontario Mathematics Curriculum 2000

Student Destinations1999-2000 Cohort to Fall 2004

33% to University

19% to College

30%Leave before OSSD

18% OSSD to Work

Source: Alan King, Double Cohort Study 2005

Grade 9 Enrolment = 100%

Double Cohort Study – Phase 4Grade 11 Achievement

Grade 11 Courses

2001-2002 2002-2003 2003-2004

Functions and Relations (U) 11.4% 11.0% 9.2%Functions (U/C) 20.9% 19.7% 18.2%Personal Finance (C) 18.6% 17.3% 16.5%Math For Everyday Life (W)

17.0% 15.8% 15.3%

Marks Distribution (% Grade 11 Students 2003-2004)

0

5

10

15

20

<50 50 51-55 56-60 61-65 66-70 71-75 76-80 81-85 86-90 >90

Achievement

% S

tud

ents

MEL 3E MBF 3C MCF 3M MCR 3U ENG 3U

Grade 11 Student Achievement

Double Cohort Study: Phase 4, 2005

Marks Distribution (% Grade 12 Students 2003-2004)

0

5

10

15

20

<50 50 51-55 56-60 61-65 66-70 71-75 76-80 81-85 86-90 >90

Achievement

% S

tud

ents

MEL 4E MAP 4C MCT 4C MDM 4U MCB 4U

MGA 4U ENG 4U

Grade 12 Student Achievement

Double Cohort Study: Phase 4, 2005

PISA 2003: Indices of Student Engagement In Mathematics (15 year olds)

Significantly higher than Canadian average

Performing the same as the Canadian average

Significantly lower than Canadian average

Interest and enjoyment in mathematics

ONTARIO

NFLD, PEI, NS, NB, QU, MAN, SK, AL

BC

Belief in usefulness of mathematics

NS, QU NFLD, PEI, MAN, SK, AL

ONTARIO

NB, BC

Mathematics confidence

QU, AL NFLD, BC ONTARIO

PEI, NS, NB, MAN, SK

Perceived ability in mathematics

QU, AL NFLD, PEI, NS, NB, SK

ONTARIO

MAN, BC

Mathematics anxiety

ONTARIO NB, QU, MAN, SK, AL, BC

NFLD, PEI, NS

Double Cohort Study – Phase 4Grade 11 Enrolment

Grade 11 Courses

2001-2002 2002-2003 2003-2004

Functions and Relations (U) 34.3% 28% 26.8%Functions (U/C) 26.2% 27.4% 26.1%Personal Finance (C) 29.6% 32.8% 34.4%Math For Everyday Life (W)

10% 11.7% 12.7%

Consultation with Colleges

• Heads of Technology - Spring 2004• College Math Survey – April 2004• ACAATO consultation - June 2004• Colleges Gr 9/10 feedback – Nov 2004• Revision writing - July 2005• Feedback consultation Gr 11/12 - Nov 2005• Grade 11Consultations – Spring 2006• Revision writing – July 2006

ACAATO Recommendations 2004

• Create a clearer pathway from Grade 10 Applied to Grade 12 College Tech

• Revise Grade 11 Personal Finance course to better prepare students for Grade 12C.

• Address overlap in 11U and 11M to ensure 11M is more appropriate for students entering college tech programs.

ACAATO Recommendations 2004

• Grade 12 College Tech should be more appropriate for college bound students;

• Improve how the curriculum helps students develop concepts, basic numeric and algebraic skills and the ability to apply processes such as problem-solving, estimation and communication.

ACAATO Research 2004

Math-related Program Clusters: Applied Arts Business Health Sciences Hospitality Human Services Technology Skilled Trades

Review Process: Synthesis

Revisions address:

• Curriculum Expectations

• Equity

• Learning

• Teaching

• Assessment and Evaluation

• Learning Tools

Goals Of Revision:

• Reduce the density of the curriculum

• Provide more opportunities for students to develop and apply important life-long process skills

• Provide clearer pathways

• Incorporate more grade and destination appropriate topics and skills

Goals Of Revision:

• Enhance curriculum coherence and concept development over the grades

• Improve student achievement and graduation rates

• Improve access to higher mathematics, attitudes towards mathematics, student retention in mathematics

MATHEMATICAL PROCESSES

OVERALL/SPECIFIC EXPECTATIONS

ACHIEVEMENT CHART

SAMPLE PROBLEMS

PATHWAYS REVIEW

EXAMPLES

INTRODUCTION

42

STRANDS

SUBHEADINGS

Mathematics Grades 11 and 12

Review Process: Feedback Consultations

• feedback consultation on proposed revisions to Grades 11 and 12 occurred in the fall of 2005

Day 1 - information provided on curriculum review process and the proposed revisions in the draft Day 2 - participants share feedback on the draft of proposed revisions gathered through a consultation process within their board or organization

• information from the consultations and feedback sessions informs further revisions

Review Process: Feedback ConsultationsGrade 11: Foundations for College Math

Strengths:

• destination appropriate

• expectations clearer; examples and sample problems clarify the intended depth, breadth, and level of difficulty

• better results; expanding on topics introduced in grade 10 provides better preparation for grade 12 College course

Review Process: Feedback ConsultationsGrade 11: Foundations for College Math

Suggestions and Considerations:• more examples and “Sample Problems”• identify use of technology in more specific

places• consider impact of availability of local

technology to support implementation• students from 10 Applied without a strong

foundation may find this course challenging

Review Process: Feedback ConsultationsGrade 11: Functions and Applications

Strengths: provides good grounding for broad range of

math applications revised version is more destination

appropriate for college bound students many expectations call for investigation; the

smaller number of expectations should help support this change

clarity of expectations

Review Process: Feedback ConsultationsGrade 11: Functions and Applications

Suggestions and Considerations:• more examples and sample problems• more examples of the use of technology• should ‘radians’ have been removed?• the course is better preparation for Gr. 12C

Math for College Tech than for the Gr. 12U Data Management course

• students from 10 Applied may find the course challenging

Media Response to Revision

Public response to the proposed DRAFT Senior Mathematics

revisions focused primarily on the issue of Calculus.

Questions Raised:

• Does COMPLEX = DIFFICULT?• Does RIGOROUS = HARD ?• How many are served better by a DENSE CURRICULUM? • For whom is a DENSE CURRICULUM developmentally

appropriate?• Who is marginalized by a DENSE CURRICULUM?• What is the relationship between content density and

curriculum quality?

CONTENT DENSITY

CU

RR

ICU

LU

M

QU

AL

ITY

CONTENT DENSITY

CU

RR

ICU

LU

M

QU

AL

ITY

Review Process: Minister’s Task Force

• February 16, 2006 – Minister announces extended review and organization of the Ministry of Education’s first Curriculum Council: Task Force on Senior High School Mathematics

• February/March 2006 – Task Force Consultations

• April 2006 – Task Force submits report • June 2006 – Task Force report released

(www.edu.gov.on.ca)

Task Force Recommendation:

• That the Grade 12 courses Mathematics for Work and Everyday Life, Foundations for College Mathematics and Mathematics for College Technology be implemented essentially as currently planned.

Key Message: Curriculum

The revised curriculum is more coherent, focused on important

mathematics and well articulated across the grades.

Summary of Key Changes

• address concerns regarding an overcrowded curriculum: reduced the number of expectations (e.g., removed “Conics ” strand from Grade 11U Functions course)

• address high failure rates: (e.g., concepts developed in a more developmentally appropriate manner and link better with Grade 10)

• create clearer pathways to Grade 12 from Grades 9 and 10 Applied Mathematics courses ( e.g., revised 11U/C to articulate with both 10 Academic and 10 Applied)


• create clearer pathways for students not entering mathematics or science programs at university (e.g., created a more focused pathway through Grade 11U/C Function Applications to Grade 12);

• revise expectations to reflect a better balance between the development of procedural fluency, deeper conceptual understanding and the ability to apply key mathematical processes like problem solving, communication and reasoning.

• improve curriculum coherence (e.g., reorganized strands in college destination courses to improve concept development);


• reduce or eliminate overlap (e.g., reduced overlap between Grade 11U and Grade 11U/C mathematics courses);

• engage students in a more relevant high school learning experience by increasing emphasis on connections within mathematics and between mathematics and the real-world (e.g., stronger connections between topics related to functions from Grade 9 through to Grade 11, increased career connections in the Grade 11 workplace course)

• encourage the use of a broad range of learning tools to support meaningful student learning in mathematics (e.g., revised specific expectations to include more references to the use of technological tools like graphing technology, calculators, statistical software)

Clear Pathways (DRAFT)

Grade 9Academic

Grade 10Academic

Grade 11EWork and

Everyday Life

Grade 10Applied

Grade 9Applied

Grade 11 MFunction

Applications

Grade 11 C Foundations for

College Math

Grade 12 C Foundations for

College Math

Grade 12CCollege

Technology

Grade 12UData

Management

Grade 12UAdvanced Functions

Grade 11 UFunctions

Grade 9L.D.C.C.

Grade 10L.D.C.C.

Calculus and Vectors 12U Course

Grade 12EWork and

Everyday Life

T

Comparing Strands: Grade 11U

2000 Curriculum

Financial Applications of Sequences and Series– Arithmetic/Geometric Sequences and Series– Compound Interest and Annuity Problems– Financial Decision Making

• Trigonometric Functions– Sine Law/Cosine Law for Oblique Triangles– Understanding and Applying Radian Measure– Graphs and Equations of Sinusoidal Functions– Models of Sinusoidal Functions

• Tools for Operating and Communicating with Functions

– Polynomials/Rational Expressions and Exponential Expressions

– Inverses/Transformations/Function Notation– Mathematical Reasoning

• Loci and Conics– Loci– Equations– Solving Problems

Revised 2006 Curriculum

• Characteristics of Functions– Representing Functions– Solving Problems Involving Quadratic Functions– Determining Equivalent Algebraic Expressions

• Exponential Functions– Representing Exponential Functions– Connecting Graphs and Equations of Exponential

Functions– Solving Problems Involving Exponential

Functions

• Discrete Functions– Representing Sequences– Investigating Arithmetic and Geometric

Sequences and Series– Solving Problems Involving Financial

Applications

• Trigonometric Functions– Determining and Applying Trigonometric Ratios– Connecting Graphs and Equations of Sinusoidal

Functions.– Solving Problems Involving Sinusoidal Functions

Revision Highlights: 11U

Increased focus on:• characteristics of

functions;• transformations;• exponential functions;• discrete functions;• modelling;• rate of change;• radical expressions;• reciprocal trig identities;• periodic functions;

Decreased focus on:• conics and loci;• annuities and mortgages;• solving exponential

equations;• solving trig equations;• complex roots;• radians;• tangent function;• solving linear inequalities

Return

Comparing Strands: Grade 11C

2000 Curriculum

• Models of Exponential Growth– Nature of Exponential Growth– Mathematical Properties of Exponential

Functions– Manipulating Expressions

• Compound Interest/Annuities– Arithmetic/Geometric Sequences and

Series– Compound Interest and Annuity Problems– Effect of Compounding

• Personal Financial Decisions– Owning/Operating A Vehicle– Renting/Buying Accommodation– Designing Budgets– Making Informed Decisions– Career Opportunities


• Mathematical Models– Connecting Graphs and Equations of Quadratic

Relations– Connecting Graphs and Equations of

Exponential Relations– Solving Problems Involving Exponential

Relations

• Personal Finance– Solving Problems Involving Compound

Interest– Comparing Financial Services– Owning/Operating A Vehicle

• Geometry and Trigonometry– Representing Two-Dimensional Shapes and

Three-Dimensional Figures– Applying the Sine Law and the Cosine Law in

Acute Triangle

• Data Management– Working With One-Variable Data– Applying Probability

Revision Highlights: 11C

Increased focus on:• quadratic relations;• modelling;• exponents;• two-dimensional

shapes;• three-dimensional

figures;• sine and cosine laws;• one variable statistics;• probability;

Decreased focus on:• sequences and series;• annuities and

mortgages;• financial decision

making;• career opportunities;

Return

Comparing Strands: Grade 11E

2000 Curriculum

• Earning, Paying Taxes and Purchasing– Earning Money– Describing Forms of Taxation– Purchasing Items

• Saving, Investing and Borrowing– Calculating Simple and Compound

Interest– Understanding Saving and Investing– Understanding Borrowing

• Transportation and Travel– Understanding the Costs of Owning and

Operating a Vehicle– Understanding the Costs of Travelling

by Automobile– Comparing Travel Costs


• Earning and Purchasing– Earning– Describing Purchasing Power– Purchasing

• Saving, Investing and Borrowing– Comparing Financial Services– Saving and Investing– Borrowing

• Transportation and Travel– Owning and Operating a – Travelling by Automobile– Comparing Modes of Transportation

Revision Highlights: 11E

Increased focus on:

• connections to workplace;

• gathering and interpreting information;

Decreased focus on:

• personal income tax;

• monitoring value of investments;

Return

Comparing Strands: Grade 11M

2000 Curriculum

• Financial Applications of Sequences and Series

– Arithmetic/Geometric Sequences and Series– Compound Interest and Annuity Problems– Financial Decision Making

• Trigonometric Functions– Sine Law/Cosine Law for Oblique Triangles– Understanding and Applying Radian Measure– Graphs and Equations of Sinusoidal Functions– Models of Sinusoidal Functions

• Tools for Operating and Communicating with Functions

– Polynomials/Rational Expressions and Exponential Expressions

– Inverses/Transformations/Function Notation– Mathematical Reasoning


• Quadratic Functions– Solving Quadratic Equations– Connecting Graphs and Equations of

Quadratic Functions– Solving Problems Involving Quadratic

Functions

• Exponential Functions– Connecting Graphs and Equations of

Exponential Functions– Solving Problems Involving Exponential

Functions– Solving Financial Problems Involving

Exponential Functions

• Trigonometric Functions– Applying the Sine Law and the Cosine Law

in Acute Triangles– Connecting Graphs and Equations of Sine

Functions– Solving Problems Involving Sine

Functions

Revision Highlights: 11M

Increased focus on:• characteristics of

functions;• quadratic functions;• exponential functions;• modelling;• rate of change;• periodic functions;

Decreased focus on:• sequences and series;• rational expressions;• annuities and mortgages;• solving exponential equations;• solving trig equations;• complex roots;• radians;• cosine/tangent functions;• rational expressions;• inverse functions;• transformations;• solving linear inequalities;

Return

MP

M2DM

BF

3C

MCR3U

MFM2P

MCF3M

Grade 11M: Functions and Applications Connections to Other Courses

Concept Development: Looking at Financial Concepts

Concept Development: Looking at Functions

Revising the Expectations

• some expectations were revised by:

- combining similar expectations

- folding expectations into processes

- reducing overlap of content among expectations

- removing inappropriate expectations

• some expectations were expanded for clarity

Eliminating Redundancy

2006 REVISED CURRICULUM

Grade 11U: Functions


Grade 11M: Functions and Applications

• Understanding Functions

• Exponential Functions

• Discrete Functions

• Trigonometric Functions

• Quadratic Functions

• Exponential Functions

• Trigonometric Functions

Improving Clarity

2000 CURRICULUM

Grade 11E: Mathematics for Everyday Life

2006 REVISED CURRICULUM Grade 11E: Mathematics for Work and

Everyday Life

•calculate compound interest by using the simple-interest formula and a given spreadsheet template;

•determine, through investigation using technology, the compound interest for a given investment, using repeated calculations of simple interest for no more than six compounding periods. (Sample problem: Someone deposits $5 000 at 4% interest per annum, compounded semi-annually. How much interest accumulates in 3 years? );

Real-world Connections

2006 DRAFT REVISED CURRICULUM Grade 11M:

Functions and Applications

solve problems arising from real-life situations, given the algebraic representation of quadratic relationship (e.g., given the equation of a quadratic function representing the height of a ball over an elapsed time, answer questions that involve finding the maximum height of the ball, the length of time needed for the ball to touch the ground, and the time interval when the ball is higher than a given measurement) (Sample problem: The relationship between power dissipated in a load resistor, P (in Watts, W), electrical potential (in Volts, V), current (in amperes, A) and resistance , R (in Ohms, Ω) is described by the formula P = EI – I2R. If the electrical potential is fixed at 24 V, and the resistance is fixed at 1.5 Ω , determine graphically and algebraically the current that results in the maximum power dissipated.) < NEW >


2006 REVISED CURRICULUM Grade 11M:


collect data arising from applications that can be modelled as an exponential relation, through investigation with and without technology, from primary sources using a variety of tools (e.g., concrete materials; measurement tools such as electronic probes) or from secondary sources (e.g., web sites such as Statistics Canada, E-STAT), and graph the data (Sample problem: Collect data and graph the cooling curve representing the relationship between temperature and time for hot water cooling in a porcelain mug. Predict the shape of the cooling curve when hot water cools in an insulated mug. Test your prediction.)


National Debt (1867-2005)

0

100000

200000

300000

400000

500000

600000

700000

1 11 21 31 41 51 61 71 81 91 101 111 121 131

Year

De

bt

in M

illio

ns

Year

Debt

There was a time when some said the national debt increased exponentially. Determine if there is a domain over which the graph of the National Debt could be modelled by an exponential curve.


More Examples

2000 CURRICULUM

Grade 11E: Mathematics for Everyday Life

2006 REVISED CURRICULUM Grade 11E: Mathematics for Work and

Everyday Life

< NEW >

•describe the effects of different remuneration methods (e.g., hourly rate, overtime rate, job or project rate, commission, salary, gratuities) and remuneration schedules (e.g., weekly, biweekly, semi-monthly, monthly) on decisions related to personal spending habits (e.g., the timing of a major purchase, the scheduling of mortgage payments and other bill payments.);

Key Message: Equity

The revised curriculum supports equity by promoting excellence in

mathematics education for all students.

Equity – NCTM Perspective

• All students, regardless of their personal characteristics, backgrounds, or physical challenges, must have opportunities to study and support to learn mathematics

• All students need access each year they are in school to a coherent, challenging mathematics curriculum taught by competent and well-supported mathematics teachers.

• Too many students, especially students who are poor, not native speakers of English, disabled, female, or members of minority groups, are victims of low expectations in mathematics.

Equity – Feedback

• Revisions must meet the needs of the students entering mathematics-related university programs.

• Equal access to senior mathematics courses across the province is very important.

• The current curriculum is too dense resulting in a reduction of students engaging in senior mathematics and a decrease in the chance of success for some students.

What Factors Contribute Most To Students’ Success in Mathematics?

• active participation in meaningful mathematics;• in-depth understanding of mathematics is supported

by active involvement in mathematical modelling, problem solving and reasoning through application

• ample time to perform investigations and to revise work;

• classroom practices that encourage discussion among students and between students and teachers;

• student reflection on their work;• an appreciation of student diversity.

Ed Thoughts 2002 – Research and Best Practice.

• learning experiences that involve a range of activity from short whole-group instruction to longer times engaged in problem solving

• positive student-teacher relationships

• “user-friendly” classroom environments in which prior knowledge is identified and built upon, and where instruction is developmentally appropriate

Ed Thoughts 2002 – Research and Best Practice.

What Factors Contribute Most To Students’ Success in Mathematics?

Equity: Developmentally Appropriate

A developmentally appropriate curriculum

• is challenging but attainable for most students of a given age group preparing for a given destination

• allows enough flexibility to respond to inevitable individual variation

• is consistent with the students’ ways of

thinking and learning

(Adapted from Clements, Sarama & DiBiase, 1997)

How do Students’ Attitudes Affect Their Performance and Future Opportunities?

Students’ attitudes toward mathematics have a great effect on student achievement. • Students who enjoy mathematics tend to perform

well in their mathematics course work and are more likely to enrol in the more advanced mathematics courses.

• Students who dislike mathematics tend not to do well in these classes, and/or do not attempt the more advanced mathematics classes in secondary school.

Ed Thoughts 2002 – Research and Best Practice


Students develop positive attitudes when they

• make mathematical conjectures;

• make breakthroughs as they solve problems;

• see connections between important ideas.

Ed Thoughts 2002: Research and Best Practice


Students with a productive attitude

• find sense in mathematics,

• perceive it as both useful and worthwhile,

• believe that steady effort in learning mathematics pays off

• view themselves as effective learners and doers of mathematics.



Students experience frustration when they are not making progress towards solving a problem. Therefore, it is important that instruction provide appropriately challenging problems so students can learn and establish the norm of perseverance for successful problem solving.


Equity

Students can be considered to be “at-risk” when they are in peril of not reaching their learning potential.

CMESG Work Group

Personal Reflection

Reflection:Most students who take mathematics do not

pursue post secondary destinations that have an emphasis on mathematics. What are the important skills you believe these students should develop

through senior mathematics?

Key Message: Learning

The revised curriculum supports students learning mathematics with understanding and actively building new knowledge from

experience and prior knowledge.

We use the ideaswe already have(blue dots) toconstruct newideas (red dot).The more ideas we use and the more connectionswe make, the better we understand.

Developing UnderstandingDeveloping Understanding

John Van de Walle

Conceptual Understanding

• Conceptual understanding supports retention. When facts and procedures are learned in a connected way, they are easier to remember and use and can be reconstructed when forgotten.

Hiebert and Wearne 1996; Bruner 1960, Katona 1940

Improving Articulation Across The Grades

Academic Pathway Applied Pathway

Grade 9 •Linear Relations •Linear Relations

Grade 10 •Quadratic Relations •Modeling Linear Relations•Quadratic Relations

Draft

Grade 11

•Understanding Functions•Exponential Functions•Discrete Functions•Trigonometric Functions

•Mathematical Models–Quadratic Relations–Exponential Relations

Proposed Grade 12

(Nov 2005)

•Polynomial Functions•Trigonometric, Exponential and Logarithmic Functions•Rates of Change

Mathematical Models:–Solving Exponential Equations–Interpreting and Analyzing Graphical Representations–Interpreting and Analyzing Algebraic Representations

Improving Articulation Across The Grades

Draft Revised Gr. 11 Foundations Proposed Draft Gr. 12 C (Nov 2005)

Mathematical Models–Investigating Graphs and Equations of Quadratic Relations–Understanding Exponential Growth and Decay–Investigating Graphs and Equations of Exponential Relations

Mathematical Models–Solving Exponential Equations–Interpreting and Analyzing Graphical Representations–Interpreting and Analyzing Algebraic Representations

Personal Finance–Solving Problems Involving Compound Interest–Investing and Borrowing–Owning and Operating A Vehicle

Personal Finance–Understanding Annuities–Renting/Buying Accommodation–Designing Budgets

Measurement and Trigonometry–Representing Two-Dimensional Shapes and Three Dimensional Figures–Applying the Sine Law and the Cosine Law in Acute Triangles

Measurement and Trigonometry–Optimization Problems–Solving Problems Involving Trigonometry

Reasoning With Data–Working with One Variable Data-Applying Probability

Reasoning With Data–Two Variable Analysis–Evaluating Validity

2000 CURRICULUM

Grade 11M: Functions

2006 DRAFT REVISED CURRICULUM

Grade 11U: Functions and Applications

•define the term function;

• explain the meaning of the term function, through investigation of linear and quadratic relations using a variety of representations (i.e., tables of values, mapping diagrams, graphs, functions machines) (Sample problem: give examples of linear and quadratic relations that are functions and that are not functions using a variety of representations);

Developing Concepts Through Investigation

Representations

Graphical Representation Numerical Representation

Algebraic Representation Concrete Representation

f(x) = 2x - 1

Culminating With Solving Problems

2000 CURRICULUM

Grade 11: Mathematics of Personal Finance

2006 DRAFT REVISED CURRICULUM Grade Grade 11: Foundations for College

Mathematics

< NEW >

•solve design problems that satisfy given constraints (e.g., design a rectangular berm that would hold all the oil that could leak from a cylindrical storage tank), using physical models (e.g., built from popsicle sticks, cardboard, duct tape) or drawings (e.g., made using design software) (Sample problem: Design and construct a model boat that can carry the most pennies, using one sheet of 8 ½” x 11” card stock and no more than five popsicle sticks)

Reflection:

Balancing Conceptual and Procedural Learning

• Does the balance vary depending on the students?

• Does the balance vary depending on the course?

• Is there an order?

• Does the balance vary depending on whether the concept is new or an extension?

Personal Reflection

Balanced Activity Reflection:What does an appropriate balance mean to you

and how does this impact on your students’ long term success in senior mathematics?

Key Message: Teaching

The revised curriculum supports effective mathematics teaching that

requires understanding what students know and need to learn

and do.

Learning mathematics … requires understanding and being able to apply procedures, concepts and processes. In the twenty-first century, all students should be expected to understand and be able to apply mathematics.

NCTM, Principles and Standards, 2000.

Teaching

Mathematical Processes: Research

Mathematical proficiency, as we see it, has five components, or strands:

• procedural fluency—skill in carrying out procedures flexibly, accurately, efficiently, and appropriately

• conceptual understanding—comprehension of mathematical concepts, operations, and relations

• strategic competence—ability to formulate, represent, and solve mathematical problems

• adaptive reasoning—capacity for logical thought, reflection, explanation, and justification

• productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy.

(Kilpatrick, Swafford, &Findell, 2001)

Mathematical Processes

Problem Solving

Reasoning and Proving

Reflecting

Selecting Tools and Computational Strategies

Connecting

Representing

Communicating

Mathematical Processes:

• the Actions of Mathematics

• ways of acquiring and using the content, knowledge and skills of mathematics

• interconnected

• not New !!


“pose and solve problems related to models of sinusoidal functions drawn from a variety of applications, and communicate the solution with clarity and justification, using appropriate mathematical forms …

pp. 24, Gr 11, 1999

solve problems

communicate

mathematical forms

pose

Connecting

Representing

modelsvariety of applications

clarity and justification

Representing

Reflecting

Reasoning and Proving

Connecting

Selecting Tools and Computational Strategies

Problem Solving

Communicating


Mathematical Proficiency


Key Message: Assessment and Evaluation

The revised curriculum supports assessment for the learning of important mathematics and to furnish useful information to both teachers and students.

Assessment and Evaluation• Do overall expectations have to be

evaluated?

YES• Do all specific expectations have to be

evaluated?

NO• Do all specific expectations have to be

taught?

YES

Knowledge and Understanding

• Factual/Procedural Knowledge• Relationships (e.g. Pythagorean Relationship)• Procedural Fluency (e.g. multi-digit

computation)• Meanings of terms in mathematics (e.g.,

property, parallelogram)

• Conceptual Understanding• Reflecting an understanding of mathematical

concepts (e.g. place value, area, rate)

Thinking

Use of planning skills• understanding the problem (e.g., formulating and

interpreting the problem, making conjectures)• making a plan for solving the problem

Use of processing skills• carrying out a plan (e.g., collecting data, questioning,

testing, revising, modelling, solving, inferring, forming conclusions)

• looking back at the solution (e.g., evaluating reasonableness, making convincing arguments, reasoning, justifying, proving, reflecting)

Use of critical/creative thinking processes (e.g., problem-solving, inquiry)

Application

• Application of knowledge and skills in familiar contexts

• Transfer of knowledge and skills to new contexts

• Making connections within and between various contexts (e.g., connections between concepts, representations, and forms within mathematics; connections involving use of prior knowledge and experience; connections between mathematics, other disciplines, and the real world)

Communication

• Expression and organization of ideas and mathematical thinking using oral, visual and written forms

• Communication for different audiences and purposes in oral, visual, and written forms

• Use of conventions, vocabulary, and terminology of the discipline in oral, visual, and written forms

Key Message: Learning Tools

The revised curriculum supports the use of

technology and manipulatives as tools for teaching and

learning mathematics.

Learning Tools: Dynamic Geometry Software/Spreadsheets

2000 CURRICULUM

Grade 11M: Functions


Grade 11M:


<NEW>

•verify, through investigation using technology (e.g., dynamic geometry software, spreadsheets) the sine law and the cosine law (e.g., compare, using dynamic geometry software, the ratios of a/sin A, b/sin B and c/sin C in triangle ABC, while dragging one of the vertices);

Learning Tools:Dynamic Statistics Software/Spreadsheets

2000 CURRICULUM Grade 11C:

Personal Finance

2006 REVISED CURRICULUM Grade 11C: Foundations of

Mathematics

< NEW>

•collect one-variable data from secondary sources (e.g., internet databases) and organize and store the data using a variety of tools (e.g., spreadsheets, dynamic statistical software);

Learning Tools: Calculators and Manipulatives

2000 CURRICULUM

Grade 11C: Mathematics of Personal Finance

2006 CURRICULUM Grade 11C: Foundations of Mathematics

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

• expand and simplify, using a variety of tools (e.g., paper and pencil, algebra tiles, computer algebra systems) quadratic expressions in one-variable, involving multiplying and squaring of binomials (e.g., ½ x + 1)(3x – 2) or 5(3x – 1)2)

Learning Tools:

Cooling Curve

Learning Tools:

Fuel Consumption Calculator

Algebra Tiles: Completing the Square

Learning Tools:

a

Learning Tools:

TVM Solver:Doubling Time

Learning Tools:

Half-Life Activity

Next Steps

1

DELIVEREDCURRICULUM

InstructionalProgramIn The

Classroom

INTENDEDCURRICULUM

Ministry Curriculum

Expectations

ACHIEVEDCURRICULUM

What IsBeing

Assessed

Working Toward Alignment

Documents

Senior Mathematics Curriculum Revision 4 Supporting students and teachers by keeping Ontario’s K - 12 curriculum current and relevant College Math Project