FunctionsBased
CurriculumMath Camp 2008
Trish Byers
AnthonyAzzopardi
“Icebreaker”
• match each of the quotes in Column A with their dates in Column B
A B
FOCUS: FUNCTIONS BASED CURRICULUM
DAY ONE: CONCEPTUAL UNDERSTANDING
DAY TWO: FACTS AND PROCEDURES
DAY THREE: MATHEMATICAL PROCESSES
Why focus on functions?
Mathematical Proficiency
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
“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
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
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
Mathematical Processes:
• The actions of mathematics
• Ways of acquiring and using the content, knowledge and skills of mathematics
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.
Mathematical Processes
Reasoning and Proving
Reflecting
Representing Connecting
Selecting Tools and Strategies
Problem Solving
Communicating
Mathematical Proficiency
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
Problem Solving Model
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iinnffoorrmmaattiioonn
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uunnddeerrssttaanndd iitt bbeetttteerr
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mmaanniippuullaattiivveess ttoo iilllluussttrraattee rreessuullttss.. WWrriittee wwoorrddss aanndd ssyymmbboollss ttoo
rreepprreesseenntt sstteeppss..
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IIss tthheerree aa bbeetttteerr wwaayy?? CCoonnssiiddeerr eexxtteennssiioonnss oorr vvaarriiaattiioonnss..
CCoommmmuunniiccaattee:: CChhoooossiinngg tthhee bbeesstt ffoorrmmaatt ffoorr
ddeessccrriibbiinngg aanndd eexxppllaaiinniinngg hhooww tthhee ssoolluuttiioonn wwaass rreeaacchheedd
Developing a Broader Range of Skills and Strategies
“When the only tool you have is a hammer, every problem looks
like a nail.”Maslow
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
Give me a fish and you feed me for a day.
Teach me to fish and you feed me for life.
Chinese ProverbChinese Proverb
Communication
• THINK
• TALK
• WRITE
Communication
Problem:
Expand (a + b)3
Answer:
( a + b ) 3
Reasoning and Proving
If a 7-11 is open 24 hours a day, 365 days a year…..
Why are there locks on the doors?
Reasoning and Proving
The bigger the perimeter, the bigger the area. Do you agree? Explain.
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
• 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
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
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
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
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?
Overall Expectations
SpecificExpectations
SpecificExpectations
SpecificExpectations
SpecificExpectations
SpecificExpectations
SpecificExpectations
SpecificExpectations
EVALUATE
ProfessionalJudgement
TEACH AND ASSESS
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
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.
How can we connect the mathematical processes with the
four categories of the achievement chart in a balanced way?
Thinking
ApplicationKnowledge/Understanding
Communication
SCIENCE
The Achievement Chart
ARTS
SOCIALSTUDIES
MATHEMATICS
PHYSICALEDUCATION
LANGUAGEARTS
Knowledge and Understanding
Thinking
Communication
Application
Mathematical Concepts, Facts and
Procedures
KNOWING
Mathematical
Processes
DOING
CURRICULUM
EXPECTATIONS
ASSESSMENT
CATEGORIES
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
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