Transcript
Page 1: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

FunctionsBased

CurriculumMath Camp 2008

Page 2: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Trish Byers

AnthonyAzzopardi

Page 3: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

“Icebreaker”

• match each of the quotes in Column A with their dates in Column B

A B

Page 4: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

FOCUS: FUNCTIONS BASED CURRICULUM

DAY ONE: CONCEPTUAL UNDERSTANDING

DAY TWO: FACTS AND PROCEDURES

DAY THREE: MATHEMATICAL PROCESSES

Why focus on functions?

Page 5: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Mathematical Proficiency

Page 6: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 7: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 8: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

“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

Page 9: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Graphical Representation Numerical Representation

Algebraic Representation

Concrete Representation

f(x) = 2x - 1

Teaching: Multiple Representations

Page 10: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 11: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Use the graphs of and h(x) = 5

to verify your solution for

1

x + 1=f(x)

Multiple Representations

1

x + 1< 5

Page 12: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 13: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 14: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Should mathematics be taught the same way as line dancing?

Page 15: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 16: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 17: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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.

Page 18: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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”

Page 19: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Teaching

The problem is no longer just teaching better mathematics.

It is teaching mathematics better.

Adding It Up: National Research Council - 2001

Page 20: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Underlying Principles for Revision

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

Page 21: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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.

Page 22: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Learning Activity: FunctionsLEARNING ACTIVITY: FUNCTIONS

Rel

atio

n

Nu

mer

ical

R

epre

sen

tati

on

(e.g

., F

init

e D

iffe

ren

ces)

Gra

ph

ical

R

epre

sen

tati

on

(e.g

., Z

eros

of

Fu

nct

ion

)

Alg

ebra

ic

Rep

rese

nta

tion

(e

.g.,

Sol

vin

g

Eq

uat

ion

s)

Con

cep

t of

F

un

ctio

n D

omai

n

and

R

ange

Tra

nsf

orm

atio

ns

Inve

rse

Linear

Quadratic

Exponential

Trig

Polynomial

Rational

Page 23: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 24: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 25: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 26: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 27: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 28: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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:

Page 29: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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:

Page 30: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

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

Page 31: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi
Page 32: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi
Page 33: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi
Page 34: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Mathematical Processes:

• The actions of mathematics

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

Page 35: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Mathematical Processes and the MathematicianMathematicians, in short, are typically somewhat lost and bewildered most of the time that they are working on a problem. Once they find solutions, they also have the task of checking that their ideas really work, and that of writing them up, but these are routine, unless (as often happens) they uncover minor errors and imperfections that produce more fog and require more work. What mathematicians write, however, bears little resemblance to what they do: they are like people lost in mazes who only describe their escape routes never their travails inside. - Dan J. Kleitman

Professor at M.I.T.

Page 36: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Mathematical Processes

Reasoning and Proving

Reflecting

Representing Connecting

Selecting Tools and Strategies

Problem Solving

Communicating

Page 37: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Mathematical Proficiency

Page 38: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Reasoning and Proving

Reflecting

Representing Connecting

Selecting Tools and Strategies

Problem Solving

Communicating

CONCEPTS

CONCEPTS

SKILLS

SKILLS

SKILLS

CONCEPTS

FACTS

FACTS

FACTS

PRIOR KNOWLEDGE AND UNDERSTANDING

Page 39: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Problem Solving Model

NNdd UUnnddeerrssttaannddiinngg tthhee PPrroobblleemm RReerreeaaddiinngg aanndd rreessttaattiinngg pprroobblleemm IIddeennttiiffyyiinngg ggiivveenn aanndd nneeeeddeedd

iinnffoorrmmaattiioonn

CCoommmmuunniiccaattee:: TTaallkkiinngg aabboouutt tthhee pprroobblleemm ttoo

uunnddeerrssttaanndd iitt bbeetttteerr

MMaakkee aa PPllaann CCoonnssiiddeerr ppoossssiibbllee ssttrraatteeggiieess SSeelleecctt ssttrraatteeggyy oorr bblleenndd ssttrraatteeggiieess..

CCoommmmuunniiccaattee:: TTaallkkiinngg ttoo ccllaarriiffyy mmeetthhooddss LLiisstteenniinngg ttoo iiddeeaass ooff ootthheerrss

CCaarrrryy OOuutt tthhee PPllaann EExxeeccuuttee ssttrraatteeggyy ccaallccuullaattiioonnss MMoonniittoorr ssuucccceessss RReevviissee aass nneecceessssaarryy

CCoommmmuunniiccaattee:: DDrraawwiinngg ppiiccttuurreess,, uussiinngg

mmaanniippuullaattiivveess ttoo iilllluussttrraattee rreessuullttss.. WWrriittee wwoorrddss aanndd ssyymmbboollss ttoo

rreepprreesseenntt sstteeppss..

LLooookk BBaacckk CChheecckk rreeaassoonnaabblleenneessss ooff aannsswweerr RReevviieeww mmeetthhooddss:: DDooeess iitt mmaakkee sseennssee??

IIss tthheerree aa bbeetttteerr wwaayy?? CCoonnssiiddeerr eexxtteennssiioonnss oorr vvaarriiaattiioonnss..

CCoommmmuunniiccaattee:: CChhoooossiinngg tthhee bbeesstt ffoorrmmaatt ffoorr

ddeessccrriibbiinngg aanndd eexxppllaaiinniinngg hhooww tthhee ssoolluuttiioonn wwaass rreeaacchheedd

Page 40: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Developing a Broader Range of Skills and Strategies

“When the only tool you have is a hammer, every problem looks

like a nail.”Maslow

Page 41: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Problem Solving Strategies

• Guess, check, revise• Draw a picture• Act out the problem• Use manipulatives• Choose an operation.• Solve a simpler

problem.• Use technology

• Make a table• Look for a pattern• Make an organised

list• Write an equation• Use logical reasoning• Work backwards

NCTM 1987

Page 42: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Give me a fish and you feed me for a day.

Teach me to fish and you feed me for life.

Chinese ProverbChinese Proverb

Page 43: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Communication

• THINK

• TALK

• WRITE

Page 44: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Communication

Problem:

Expand (a + b)3

Answer:

( a + b ) 3

Page 45: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Reasoning and Proving

If a 7-11 is open 24 hours a day, 365 days a year…..

Why are there locks on the doors?

Page 46: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Reasoning and Proving

The bigger the perimeter, the bigger the area. Do you agree? Explain.

Page 47: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

DECK

=

=

І

І

Minds On: Deck Problem

COTTAGE

You have been hired to build a deck attached the second floor of a cottage using 48 prefabricated 1m x 1m ………

……………………………

http://www.beachside-bb.nf.ca/Accomdations.htm

Page 48: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

• Divide into groups of three to solve the problem.

• Two of your team solve the problem while the third person generates a list of “look fors” by observing and recording behaviours that serve as evidence the Mathematical Processes are being applied.

• Think about how students in “your” course might solve this problem.

• With a new “observer”, determine a second solution using different tools and strategies

Procedure:

Minds On: Deck Problem

Page 49: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

DECK

=

=

І

І

COTTAGE

You have been hired to build a deck attached to the second floor of a cottage using 48 prefabricated 1m x 1m sections. Determine the dimensions of at least 2 decks that can be built in the configuration shown.

http://www.beachside-bb.nf.ca/Accomdations.htm

Minds On: Deck Problem

Page 50: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Graphical Representation

ShortEdge

Long Edge

1

2

3

4

6

8

24.5

13

9.5

8

7

7

Numerical Representation

AlgebraicRepresentation

Concrete Representation

2xy – x2 = 48x

xy

2

48 2

Cottage

Page 51: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Deck Problem - Tiles

Cottage

Perfect SquareNumber

Even Number of Tiles Remaining

48 – 12 = 47

48 – 22 = 44

48 – 32 = 37

48 – 42 = 32

48 – 52 = 23

48 – 62 = 12

Page 52: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Deck Problem – Algebraic

x2

x48y

2

x

x

xy

22

48 2

2

24 x

xy x must be even and x

must divide evenly into 24.

1

2

3

4

6

8

12

24

x ≠ 0Can x = 8?

Can x = 12?

Can x = 24?

Page 53: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Overall Expectations

SpecificExpectations

SpecificExpectations

SpecificExpectations

SpecificExpectations

SpecificExpectations

SpecificExpectations

SpecificExpectations

EVALUATE

ProfessionalJudgement

TEACH AND ASSESS

Page 54: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

What do I want them to learn?

How will I know they have learned it?

How will I design instruction for learning?

Overall and SpecificExpectationsEssential•enduring

Achievement ChartCategoriesFramework•Reference Point

EvaluationMeasure learning at certain checkpoints during

the learning and near the end

Instructional StrategiesAnd ResourcesScaffoldingDifferentiationAssessment strategies and tools

Assessment for LearningOngoing monitoring of stu-dent progress•Sharing goals & criteria•Feedback, questioning•Peer and self-assessment•Formative use of testsAdjusting instruction

How will I respond to students who aren’t making progress?

Planning

Page 55: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Assessment and Evaluation:The following graphs are combinations of the functions: f(x) = sin x, and g(x) = x. State the combination of f(x) and g(x) (i.e., addition, subtraction, multiplication, division) that has been used to generate each graph. Justify your answer by making reference to the key features of functions.

Page 56: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

How can we connect the mathematical processes with the

four categories of the achievement chart in a balanced way?

Thinking

ApplicationKnowledge/Understanding

Communication

Page 57: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

SCIENCE

The Achievement Chart

ARTS

SOCIALSTUDIES

MATHEMATICS

PHYSICALEDUCATION

LANGUAGEARTS

Knowledge and Understanding

Thinking

Communication

Application

Page 58: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Mathematical Concepts, Facts and

Procedures

KNOWING

Mathematical

Processes

DOING

CURRICULUM

EXPECTATIONS

ASSESSMENT

CATEGORIES

Page 59: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

Making Connections

Reasoning and Proving

Thinking

Problem Solving

Knowledge and Understanding

Reflecting

Communication

Application

RepresentingCommunicating

Selecting Tools and Strategies Connecting

Procedural Knowledge Conceptual Understanding

Mathematical Processes

Page 60: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi

New Tricks?

High above the hushed crowd, Rex tried to remain focused.Still, he couldn’t shake one nagging thought;

He was an old dog and this was a new trick

Page 61: Functions Based Curriculum Math Camp 2008. Trish Byers Anthony Azzopardi