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
Revised Prerequisite Chart
Grade 9
Principles
Academic
MPM1D
Grade 10
Principles
Academic
MPM2D
T
Grade 9
Foundations
Applied
MFM1P
Grade 10
Foundations
Applied
MFM2P
Grade 9
Grade 10
LDCC
LDCC
Grade 11 U
Functions
MCR3U
Grade 11 U/C
Functions and
Applications
MCF3M
Grade 11 C
Foundations for
College Mathematics
MBF3C
Grade 11 Mathematics
for Work and
Everyday Life
MEL3E
Grade 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
Principles Underlying Curriculum Revision
•Curriculum Expectations
•Learning
•Teaching
•Assessment/Evaluation
•Learning Tools
•Equity
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
AND
1 001 = 13 x 11 x 7
Why is it so
important for
us to improve
our teaching of
mathematics?
Underlying Principles for Revision
• 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
Identify 3 key points from your article
segment.
What is one idea from the classroom that
reminds you of these ideas?
Underlying Principle for Revision
• 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;
Mathematical Proficiency
Mathematical Proficiency
Mathematical Proficiency
Representing
Problem Solving
Reflecting
Communicating
Reasoning and Proving
Connecting
Selecting Tools and
Computational Strategies
Mathematical Processes
Mathematical Proficiency
Teaching
Mathematical
Expert
Pedagogical
Expert
Teachers use
• strong subject/discipline content knowledge
• good instructional skills
• strong pedagogical content knowledge
Teacher
Student
Curriculum
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 with
length 5 units and width 3 units?
Teacher:
What is the perimeter of this
rectangle?
Student:
16
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
Interest and
enjoyment in
mathematics
Performing the
same as the
Canadian average
Significantly lower
than Canadian
average
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
Conceptual Understanding
• “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).
Frayer Model
3 Groups
Definition
Facts/Characteristics
•Grade 7/8
•Grade 9/10
•Grade 11/12
FUNCTIONS
Examples
Non Examples
Conceptual Understanding
“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).
Teaching: Multiple Representations
Graphical
Representation
Algebraic
Representation
f(x) = 2x - 1
Numerical
Representation
Concrete
Representation
Multiple Representations
MHF4U – C4.1
(x + 1)
1
< 5
x+1
1
< 5 (x + 1)
x+1
1 < 5x + 5
- 4 < 5x
-4
x >
5
Multiple Representations
1
Use the graphs of f(x) =
and h(x) = 5
x+1
1
to verify your solution for
< 5
x+1
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
Global
Wealth
Global
Population
Poor
Middle
50% 1%
49%
2%
48%
50%
Wealth
Population
0%
10%
20%
30%
40%
Wealthy
50%
Poor
60%
Middle
70%
80%
90%
100%
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
Traditional Lessons
Direct Instruction: teaching by example.
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.
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.
Linear
Quadratic
Exponential
Trig
Polynomial
Rational
Inverse
Transformations
Domain
and
Range
Concept of
Function
Algebraic
Representation
(e.g., Solving
Equations)
Graphical
Representation
(e.g., Zeros of
Function)
Numerical
Representation
(e.g., Finite
Differences)
Relation
LEARNING ACTIVITY: FUNCTIONS
Learning Activity: Functions
Grade 7 and 8
Patterning and Algebra
Grade 9 Academic
Linear Relations
Grade 9 Applied
Linear Relations
Grade 10 Academic
Quadratic Relations
Grade 10 Applied
Modelling Linear Relations
Quadratic Relations
Grade 11 Functions
Exponential, Trigonometric and
Discrete Functions
Grade 11 Foundations
Quadratic Relations
Exponential Relations
Grade 12
Advanced Functions
Exponential, Logarithmic,
Trigonometric, Polynomial, Rational
Grade 12 Foundations
Modelling Graphically
Modelling Algebraically
University Destination Transition
Functions
MCR3U
Characteristics of
Functions
Exponential
Functions
Discrete Functions
Trigonometric
Functions
Advanced Functions
MHF4U
Polynomial and
Rational Functions
Exponential and
Logarithmic Functions
Trigonometric
Functions
Characteristics of
Functions
College Destination Transition
Functions and
Applications
MCF3M
Mathematics for College
Technology
MCT4C
Quadratic Functions
Exponential Functions
Exponential
Functions
Trigonometric
Functions
Polynomial Functions
Trigonometric
Functions
Applications of
Geometry
College Destination Transition
Foundations for
Foundations for
College Mathematics College Mathematics
MBF3C
MAP4C
Mathematical Models Mathematical Models
Personal Finance
Personal Finance
Geometry and
Trigonometry
Data Management
Geometry and
Trigonometry
Data Management
Workplace Destination Transition
Mathematics for Work Mathematics for Work
and Everyday Life
and Everyday Life
MEL3E
MEL4E
Earning and
Reasoning With Data
Purchasing
Saving, Investing and Personal Finance
Borrowing
Transportation and
Travel
Applications of
Measurement
Links to Post Secondary Destinations:
UNIVERSITY DESTINATIONS:
Grade 12 U
Calculus and Vectors
MCV4U
University Mathematics, Engineering, Economics, Science,
Computer Science, some Business Programs and
Education – Secondary Mathematics
Grade 12 U
Advanced Functions
MHF4U
University Kinesiology, Social Sciences, Programs and some
Mathematics, Health Science, some Business Interdisciplinary
Programs and Education – Elementary Teaching
Grade 12 U
Mathematics of Data
Management
MDM4U
Some University Applied Linguistics, Social Sciences,
Child and Youth Studies, Psychology, Accounting,
Finance, Business, Forestry, Science, Arts,
Links to Post Secondary Destinations:
COLLEGE DESTINATIONS:
Grade 12 C
Mathematics for
College Technology
MCT4C
Grade 12 C
Foundations for
College Mathematics
MAP4C
College Biotechnology, Engineering Technology (e.g.
Chemical, Computer), some Technician Programs
General Arts and Science, Business, Human
Resources, some Technician and Health Science
Programs,
WORKPLACE DESTINATIONS:
Grade 12
Mathematics for
Work and Everyday
Life
MEL4E
Steamfitters, Pipefitters, Sheet Metal Worker,
Cabinetmakers, Carpenters, Foundry Workers,
Construction Millwrights and some Mechanics,
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
Make a set of
CO-ORDINATES
Learning Tools
Graphing Functions Using Sketchpad
Revisiting the Cube Graphically Using Winplot
N0 = (n – 2)3
f(x) = (x – 2)^3
N1 = 6(n – 2)2
f(x) = 6(x – 2)^2
N2 = 12(n – 2)
y = 12(x -2)
N3 = 8
f(x) = x3
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