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 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 9 Foundations Applied MFM1P Grade 11 Mathematics for Work and Everyday Life MEL3E Grade 9 LDCC Grade 10 Principles Academic MPM2D Grade 10 Foundations Applied MFM2P Grade 9 Principles Academic MPM1D T

Principles Underlying Curriculum Revision Learning Teaching Assessment/Evaluation Learning Tools Equity CurriculumExpectations 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 ) Is your number lucky or unlucky?

Do our students see mathematics as meaningful? magical? both?

“Icebreaker” = 3x x x x x10 + 6x = 3x x x x10 + 6x x = 3 x ( ) + 4 x ( ) + 6 x ( )

“Icebreaker” = 3 x ( ) + 4 x (10 010) + 6 x (1 001) = (3 x x100) + (4 x x 10) + (6 x x 1) = x (3x x10 + 6x1) = x 346 AND = 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

Representing Reflecting Reasoning and Proving Connecting Selecting Tools and Computational Strategies Problem Solving Communicating Mathematical Processes Mathematical Proficiency

Teaching Mathematical Expert Pedagogical Expert

Teachers use strong subject/discipline content knowledge good instructional skills strong pedagogical content knowledge Curriculum Teacher 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? Student: 16 Teacher: What is the perimeter of this rectangle?

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, ALNFLD, BCONTARIO PEI, NS, NB, MAN, SK Perceived ability in mathematics QU, AL NFLD, PEI, NS, NB, SK ONTARIO MAN, BC Mathematics anxiety ONTARIONB, 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

DefinitionFacts/Characteristics ExamplesNon 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) 1 < 5x < 5x x > 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)……. WealthyPoorMiddle Global Wealth 50% Global Population 2% 50% 1% 48% 49%

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 $ 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 Lessons Direct 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

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: Functions

Grade 9 Academic Linear Relations Grade 10 Academic Quadratic Relations Grade 11 Functions Exponential, Trigonometric and Discrete Functions Grade 12 Advanced Functions Exponential, Logarithmic, Trigonometric, Polynomial, Rational Grade 9 Applied Linear Relations Grade 10 Applied Modelling Linear Relations Quadratic Relations Grade 11 Foundations Quadratic Relations Exponential Relations Grade 12 Foundations Modelling Graphically Modelling Algebraically Grade 7 and 8 Patterning and Algebra

Functions MCR3U Advanced Functions MHF4U Characteristics of Functions Polynomial and Rational Functions Exponential Functions Exponential and Logarithmic Functions Discrete FunctionsTrigonometric Functions Characteristics of Functions University Destination Transition

Functions and Applications MCF3M Mathematics for College Technology MCT4C Quadratic FunctionsExponential Functions Polynomial Functions Trigonometric Functions Applications of Geometry College Destination Transition

Foundations for College Mathematics MBF3C Foundations for College Mathematics MAP4C Mathematical Models Personal Finance Geometry and Trigonometry 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 N 1 = 6(n – 2) 2 N 3 = 8 N 0 = (n – 2) 3 N 2 = 12(n – 2) f(x) = x 3 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