FunctionsBased

CurriculumMath Camp 2008

Trish Byers

AnthonyAzzopardi

FOCUS: FUNCTIONS BASED CURRICULUM

DAY ONE: CONCEPTUAL UNDERSTANDING

DAY TWO: FACTS AND PROCEDURES

DAY THREE: MATHEMATICAL PROCESSES

Revised Prerequisite ChartGrade 12 U

Calculus and Vectors

MCV4U

Grade 12 U Advanced Functions

MHF4U

Grade 12 U Mathematics of Data

Management MDM4U

Grade 12 C Mathematics for

College Technology MCT4C

Grade 12 C Foundations for

College Mathematics MAP4C

Grade 12 Mathematics for Work and Everyday Life

MEL4E

Grade 11 U Functions MCR3U

Grade 11 U/C Functions and Applications

MCF3M

Grade 11 C Foundations for

College Mathematics MBF3C

Grade 10

LDCC

Grade 9Foundations

AppliedMFM1P

Grade 11 Mathematics for Work and Everyday Life

MEL3E

Grade 9

LDCC

Grade 10PrinciplesAcademicMPM2D

Grade 10 Foundations

AppliedMFM2P

Grade 9PrinciplesAcademicMPM1D

T

Principles Underlying Curriculum Revision

•Learning

•Teaching

•Assessment/Evaluation

•Learning Tools

•Equity

•Curriculum Expectations

Areas adapted from N.C.T.M. Principles and Standards for School Mathematics, 2000

“Icebreaker”

• Select a three digit number. (eg. 346)• Create a six digit number by

repeating the three digit number you selected. (eg. 346346)

• Is your number lucky or unlucky?

Do our students see mathematics as

•meaningful? •magical?•both?

“Icebreaker”

• 346346 = 3x100 000 + 4x10 000 +6x1 000 + 3x100 + 4x10 + 6x1

• 346346 = 3x100 000 + 3x100 + 4x10 000 + 4x10

+ 6x1 000 + 6x1• 346346 = 3 x (100 000 + 100)

+ 4 x (10 000 + 10)+ 6 x (1 000 + 1)

“Icebreaker”

•346346 = 3 x (100 100)+ 4 x (10 010)+ 6 x (1 001)

•346346 = (3 x 1 001 x100) + (4 x 1 001 x 10) + (6 x 1 001 x 1)•346346 = 1 001 x (3x100 + 4x10 + 6x1)•346346 = 1 001 x 346

AND1 001 = 13 x 11 x 7

Why is it so important for us to improve

our teaching of mathematics?

• Equity focuses on meeting the diverse learning needs of students and promotes excellence for all by – ensuring curriculum expectations are grade and

destination appropriate,– by providing access to Grade 12 mathematics courses

in a variety of ways. – supporting a variety of teaching and learning

strategies

Underlying Principles for Revision

Identify 3 key points from your article segment.What is one idea from the classroom that reminds you of these ideas?

• Effective teaching of mathematics requires that the teacher understand the mathematical concepts, procedures, and processes that students need to learn and use a variety of instructional strategies to support meaningful learning;

Underlying Principle for Revision

Mathematical Proficiency

Mathematical Proficiency

Representing

Reflecting

Reasoning and Proving

Connecting

Selecting Tools and Computational Strategies

Problem Solving

Communicating

Mathematical Processes

Mathematical Proficiency

Mathematical Proficiency

Teaching

Mathematical Expert

Pedagogical Expert

Teachers use• strong subject/discipline content knowledge• good instructional skills• strong pedagogical content knowledge

Curriculum

Teacher

Student

Student

Pedagogical Content Knowledge

• Applying subject knowledge effectively, using concepts in ways that make sense to students

Teacher:

What is the area of a rectangle withlength 5 units and width 3 units?

Student:

16

Teacher:

What is the perimeter of thisrectangle?

Pedagogical Content Knowledge

• Applying subject knowledge effectively, using concepts in ways that make sense to student

Teacher:

What is the sin 30° + sin 60° ? Student:

sin 90°

Teacher:

Is f(x) + f(y) always equal to f(x+y)?

A Problem Solving Moment

Problem:What is the sin 50° ?

Answer:Wrinkles, Grey Hair, Memory Loss

Teaching: Student Engagement

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

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

gains-camppp.wikispaces.com

• “The concept of function is central to understanding mathematics, yet students’ understanding of functions appears either to be too narrowly focused or to include erroneous assumptions”

(Clement, 2001, p. 747).

Conceptual Understanding

Definition Facts/Characteristics

Examples Non Examples

FUNCTIONS

Frayer Model

3 Groups

•Grade 7/8•Grade 9/10•Grade 11/12

“Conceptual understanding within the area of functions involves the ability to translate among the different representations, table, graph, symbolic, or real-world situation of a function” (O’Callaghan, 1998).

Conceptual Understanding

Graphical Representation Numerical Representation

Algebraic Representation

Concrete Representation

f(x) = 2x - 1

Teaching: Multiple Representations

Multiple Representations

1

x + 1< 5

1

x + 1< 5(x + 1) (x + 1)

1 < 5x + 5

- 4 < 5x

x > -4 5

MHF4U – C4.1

Use the graphs of and h(x) = 5

to verify your solution for

1

x + 1=f(x)

Multiple Representations

1

x + 1< 5

Real World Applications MAP4C: D2.3 interpret statistics presented in the

media (e.g., the U.N.’s finding that 2% of the world’s population has more than half the world’s wealth, whereas half the world’s population has only 1% of the world’s wealth)…….

Wealthy Poor Middle

Global Wealth 50%Global Population 2% 50%

1%

48%

49%

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Population

Wealth

Wealthy Poor Middle

Real World Applications

Classroom activities with applications to real world situations are the lessons students seem to learn from and appreciate the most.

Poverty increasing: Reports says almost 30 per cent of Toronto families live in poverty.

• The report defines poverty as a family whose after-tax income is 50 percent below the median in their community, taking family size into consideration.

• In Toronto, a two-parent family with two children living on less than $27 500 is considered poor.

METRO NEWS November 26, 2007

Should mathematics be taught the same way as line dancing?

A Vision of Teaching Mathematics

• Classrooms become mathematical communities rather than a collection of individuals

• Logic and mathematical evidence provide verification rather than the teacher as the sole authority for right answers

• Mathematical reasoning becomes more important than memorization of procedures.

NCTM 1989

A Vision of Teaching Mathematics

• Focus on conjecturing, inventing and problem solving rather than merely finding correct answers.

• Presenting mathematics by connecting its ideas and its applications and moving away from just treating mathematics as a body of isolated concepts and skills.

NCTM 1989

The “NEW” Three Part Lesson.

•Teaching through exploration and investigation:•Before: Present a problem/task and ensure students understand the expectations.•During: Let students use their own ideas. Listen, provide hints and assess.•After: Engage class in productive discourse so that thinking does not stop when the problem is solved.

Traditional LessonsDirect Instruction: teaching by example.

Teaching:

Investigation

Direct Instruction

“ Effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well”

Teaching

The problem is no longer just teaching better mathematics.

It is teaching mathematics better.

Adding It Up: National Research Council - 2001

Underlying Principles for Revision

• Curriculum expectations must be coherent, focused and well articulated across the grades;

Identifying Key Ideas about Functions

• Same groups as Frayer Model Activity• Using the Ontario Curriculum, identify

key ideas about functions.• Describe the key ideas using 1 – 3 words.• Record each idea in a cloud bubble on

chart paper.

Learning Activity: FunctionsLEARNING ACTIVITY: FUNCTIONS

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R

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(e.g

., F

init

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Gra

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R

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(e.g

., Z

eros

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Fu

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)

Alg

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Rep

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(e

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Sol

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Eq

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Con

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F

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R

ange

Tra

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Inve

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Linear

Quadratic

Exponential

Trig

Polynomial

Rational

Grade 9 AcademicLinear Relations

Grade 10 AcademicQuadratic Relations

Grade 11 FunctionsExponential, Trigonometric and

Discrete Functions

Grade 12 Advanced Functions

Exponential, Logarithmic, Trigonometric, Polynomial, Rational

Grade 9 AppliedLinear Relations

Grade 10 AppliedModelling Linear Relations

Quadratic Relations

Grade 11 FoundationsQuadratic Relations

Exponential Relations

Grade 12 FoundationsModelling Graphically

Modelling Algebraically

Grade 7 and 8Patterning and Algebra

Functions MCR3U

Advanced Functions MHF4U

Characteristics of Functions

Polynomial and Rational Functions

Exponential Functions

Exponential and Logarithmic Functions

Discrete Functions Trigonometric Functions

Trigonometric Functions

Characteristics of Functions

University Destination Transition

Functions and Applications

MCF3M

Mathematics for College Technology

MCT4C

Quadratic Functions Exponential Functions

Exponential Functions

Polynomial Functions

Trigonometric Functions

Trigonometric Functions

Applications of Geometry

College Destination Transition

Foundations for College Mathematics

MBF3C

Foundations for College Mathematics

MAP4C

Mathematical Models Mathematical Models

Personal Finance Personal Finance

Geometry and Trigonometry

Geometry and Trigonometry

Data Management Data Management

College Destination Transition

Mathematics for Work and Everyday Life

MEL3E

Mathematics for Work and Everyday Life

MEL4E

Earning and Purchasing

Reasoning With Data

Saving, Investing and Borrowing

Personal Finance

Transportation and Travel

Applications of Measurement

Workplace Destination Transition

Grade 12 U Calculus and Vectors

MCV4U

Grade 12 U Advanced Functions

MHF4U

Grade 12 U Mathematics of Data

Management MDM4U

University Mathematics, Engineering, Economics, Science, Computer Science, some Business Programs and Education – Secondary Mathematics

University Kinesiology, Social Sciences, Programs and some Mathematics, Health Science, some Business Interdisciplinary Programs and Education – Elementary Teaching

Some University Applied Linguistics, Social Sciences, Child and Youth Studies, Psychology, Accounting, Finance, Business, Forestry, Science, Arts,

Links to Post Secondary Destinations:

UNIVERSITY DESTINATIONS:

Grade 12 C Mathematics for

College Technology MCT4C

Grade 12 C Foundations for

College Mathematics MAP4C

Grade 12 Mathematics for

Work and Everyday Life

MEL4E

College Biotechnology, Engineering Technology (e.g. Chemical, Computer), some Technician Programs

General Arts and Science, Business, Human Resources, some Technician and Health Science Programs,

Steamfitters, Pipefitters, Sheet Metal Worker, Cabinetmakers, Carpenters, Foundry Workers, Construction Millwrights and some Mechanics,

Links to Post Secondary Destinations:COLLEGE DESTINATIONS:

WORKPLACE DESTINATIONS:

Concept Maps• Groups of three with a representative from

7/8, 9/10 and 11/12• Use the key ideas about functions generated

earlier to build a concept map.

INPUT OUTPUT

CO-ORDINATES

Make a set of

Learning Tools

Graphing Functions Using Sketchpad

Revisiting the Cube Graphically Using Winplot

N1 = 6(n – 2)2

N3 = 8

N0 = (n – 2)3

N2 = 12(n – 2)

f(x) = x3

f(x) = (x – 2)^3

f(x) = 6(x – 2)^2

y = 12(x -2)

y = 8

Creating Graphical Models Using Winplot

•Inputting data points from Excel

•Sliders and Transformations

•Use data from investigations and model with Winplot

•Cublink Activity - Intermediate

•Winplot Activity Sheet - Senior