functions based curriculum
DESCRIPTION
Functions Based Curriculum. Math Camp 2008. Trish Byers. WELCOME BACK!. Anthony Azzopardi. FOCUS: FUNCTIONS BASED CURRICULUM DAY ONE: CONCEPTUAL UNDERSTANDING DAY TWO: FACTS AND PROCEDURES DAY THREE: MATHEMATICAL PROCESSES. Grade 12 U Calculus and Vectors MCV4U. - PowerPoint PPT PresentationTRANSCRIPT
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
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
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