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

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 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

“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

Mathematical Processes: The actions of mathematics Ways of acquiring and using the content, knowledge and skills of mathematics

Mathematical Processes and the Mathematician Mathematicians, 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 RepresentingConnecting Selecting Tools and Strategies Problem Solving Communicating

Mathematical Proficiency

Reasoning and Proving Reflecting Representing Connecting Selecting Tools and Strategies Problem Solving Communicating CONCEPTS SKILLS CONCEPTS FACTS PRIOR KNOWLEDGE AND UNDERSTANDING

Problem Solving Model

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 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 ……… ……………………………

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. Minds On: Deck Problem

Graphical Representation Short Edge Long Edge Numerical Representation Algebraic Representation Concrete Representation 2xy – x 2 = 48 Cottage

Deck Problem - Tiles Cottage Perfect Square Number Even Number of Tiles Remaining 48 – 1 2 = – 2 2 = – 3 2 = – 4 2 = – 5 2 = – 6 2 = 12

Deck Problem – Algebraic x must be even and x must divide evenly into x ≠ 0 Can x = 8? Can x = 12? Can x = 24?

Overall Expectations Specific Expectations Specific Expectations Specific Expectations Specific Expectations Specific Expectations Specific Expectations Specific Expectations EVALUATE Professional Judgement 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 Specific Expectations Essential enduring Achievement Chart Categories Framework Reference Point Evaluation Measure learning at certain checkpoints during the learning and near the end Instructional Strategies And Resources Scaffolding Differentiation Assessment strategies and tools Assessment for Learning Ongoing monitoring of stu- dent progress Sharing goals & criteria Feedback, questioning Peer and self-assessment Formative use of tests Adjusting 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 SOCIAL STUDIES MATHEMATICS PHYSICAL EDUCATION LANGUAGE ARTS 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 StrategiesConnecting Procedural KnowledgeConceptual 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