Math 3206

MathematicsMathematics 3206

GOVERNMENTOFNEWFOUNDLANDAND LABRADOR

DivisionofProgramDevelopment

MATHEMATICS 3206 CURRICULUM GUIDE i

Acknowledgments

The departments of education of New Brunswick, Newfoundland andLabrador, Nova Scotia, and Prince Edward Island gratefully acknowledgethe contributions of the following groups and individuals toward thedevelopment of this Mathematics 3206 mathematics curriculum guide.

• The regional Mathematics Curriculum Committee; current and pastrepresentatives include the following:

New Brunswick

Greta Gilmore, Mathematics Teacher,Belleisle Regional High School

John Hildebrand, Mathematics Consultant,Department of Education

Pierre Plourde, Mathematics Teacher,St. Mary’s Academy

Nova Scotia

Richard MacKinnon, Mathematics Consultant,Department of Education

Lynn Evans Phillips, Mathematics Teacher,Park View Education Centre

Newfoundland and Labrador

Sadie May, Distance Education Coordinator for MathematicsDepartment of Education

Patricia Maxwell, Program Development Specialist,Department of Education

Prince Edward Island

Elaine Somerville, Mathematics/Science Consultant,Department of Education

• The Provincial Curriculum Working Group, comprising teachersand other educators in Nova Scotia, which served as lead provincein drafting and revising the document.

• The pilot teachers and other educators and stakeholders acrossAtlantic Canada who contributed to the development of theMathematics 3206 curriculum guide.

ACKNOWLEDGMENTS

MATHEMATICS 3206 CURRICULUM GUIDEii

ACKNOWLEDGMENTS

MATHEMATICS 3206 CURRICULUM GUIDE iii

Contents

Background ................................................................................ 1

Rationale .................................................................................... 2

Meeting the Needs of All Learners ................................................. 4

Career Connections ..................................................................... 4

Curriculum

Outcomes

Curriculum Outcomes Framework ................................................ 5

Essential Graduation Learnings ..................................................... 5

General Curriculum Outcomes ...................................................... 6

Key-Stage Curriculum Outcomes .................................................. 7

Specific Curriculum Outcomes ..................................................... 7

Linking SCOs and KSCOs for Mathematics 3206 ........................... 8

Program Design

and Components

Program Organization ................................................................ 25

Content Organization ................................................................ 27

Learning and Teaching Mathematics ............................................. 30

Integrating Technology ............................................................... 31

Learning Resources .................................................................... 32

Assessing and Evaluating Student Learning ................................... 33

Course

Organization

Course Design .......................................................................... 37

The Two-Page Spread ................................................................. 37

Unit 1: Patterns ......................................................................... 39

Unit 2: Quadratics ..................................................................... 51

Unit 3: Exponential Growth ....................................................... 63

Unit 4: Circle ........................................................................... 81

Unit 5: Probability .................................................................... 97

Introduction

CONTENTS

MATHEMATICS 3206 CURRICULUM GUIDEiv

CONTENTS

MATHEMATICS 3206 CURRICULUM GUIDE 1

INTRODUCTION

Introduction

The mathematics curriculum for Atlantic Canada has been written in aneffort to align the outcomes for student learning in mathematics withthe recommendations of Curriculum and Evaluation Standards for SchoolMathematics (National Council of Teachers of Mathematics, 1989).This document identifies the primary goal for all students to be theattainment of mathematical power—the ability to make mathematicalconnections, to reason logically, to communicate and applymathematics effectively in problem situations. Since the late 1980s,several influential publications have affirmed this goal. In addition toCurriculum and Evaluation Standards for School Mathematics, theseinclude two publications from the Mathematical Sciences EducationBoard—Everybody Counts: A Report to the Nation on the Future ofMathematics Education (1989) and Reshaping School Mathematics(1990). In addition to these publications, the National Council ofTeachers of Mathematics (NCTM) published the companion standardsdocuments Professional Standards for Teaching Mathematics in 1991 andAssessment Standards for School Mathematics in 1995. In April 2000, theNCTM published its newest document, Principles and Standards forSchool Mathematics, a revision, rewriting, and restatement of the 1989Curriculum and Evaluation Standards for School Mathematics.

Foundation for the Atlantic Canada Mathematics Curriculum (1996)firmly established Curriculum and Evaluation Standards for SchoolMathematics (NCTM 1989) as a guiding beacon for pursuing thisvision, a vision that fosters the development of mathematically literatestudents. Curriculum design has been motivated by a desire to ensurethat students benefit from world-class curriculum and instruction inmathematics as a significant part of their school learning experience.More and more, students are being challenged to become problemsolvers, to understand mathematical concepts by becoming activelearners in highly interactive learning experiences. Computers andcalculators are becoming common classroom tools, and innovations inassessment of student learning (which include portfolios and open-ended questions) are being used in classrooms.

Mathematics curriculum development in this region has taken placeunder the auspices of the Atlantic Provinces Education Foundation(APEF), an organization sponsored and managed by the governments ofthe four Atlantic Provinces. The development process has broughttogether teachers and Department of Education officials to co-operatively plan and execute the development of curricula inmathematics, science, language arts, and some other subject areas. Eachof these curriculum efforts has been aimed at producing a program that

Background

MATHEMATICS 3206 CURRICULUM GUIDE2

INTRODUCTION

would ultimately support the essential graduation learnings (EGLs), alsodeveloped regionally. The essential graduation learnings, and thecontribution of the mathematics curriculum to their achievement, arepresented in the Outcomes section of the Foundation for the AtlanticCanada Mathematics Curriculum.

The mathematics foundation document provides an overview of thephilosophy and goals of mathematics curriculum, presenting broadcurriculum outcomes and addressing a variety of issues with respect tothe learning and teaching of mathematics. It describes the mathematicscurriculum in terms of a framework of outcomes—General CurriculumOutcomes (GCOs), which relate to subject strands, and Key-StageCurriculum Outcomes (KSCOs), which identify what students areexpected to learn and be able to do by the end of grades 3, 6, 9, and 12.

Each course guide builds on the structure introduced in the foundationdocument by relating Specific Curriculum Outcomes (SCOs) to eachKSCO and providing suggestions for learning experiences, instruction,assessment, and resources.

Rationale The purposes of high school mathematics are embedded in a context thatis broad and consistent with accelerating changes in today’s society—asociety that is increasingly dominated by technology and quantitativemethods. Predictions are that high school graduates in the future willchange careers at least four or five times. If we are to develop curriculumfor students who need to be flexible with respect to the workplace andcapable of lifelong learning, high school mathematics must emphasize adynamic form of literacy, and high school mathematics instruction mustmaximize the opportunity for students to achieve outcomes dealing witha broad range of topics. Experiences must be provided that encourageand enable students to gain confidence in their mathematical ability,solve mathematical problems, reason and communicate mathematically,and understand the value of mathematics.

Expectations of employers and post-secondary institutions reflect theneed for all students to understand the complexities and technologies ofcommunication, to ask questions, to assimilate unfamiliar information,and to work co-operatively. These needs are best addressed bydeveloping a curriculum that reflects the following beliefs:

€ Knowing mathematics is “doing mathematics.”Mathematics is more than just a collection of concepts and skills tobe mastered; it includes methods of investigating and reasoning,means of communication, and notions of context. Instructionalsettings and student activities should be developed and grow out ofproblem situations. This view of learning is summarized in EverybodyCounts (Mathematical Sciences Education Board, 1989).


INTRODUCTION

“In reality, no one can teach mathematics. Effective teachers are thosewho can stimulate students to learn mathematics. Educationalresearch offers compelling evidence that students learn mathematicswhen they construct their own mathematical understanding. Tounderstand what they learn, they must enact for themselves verbs thatpermeate the mathematics curriculum: ‘examine, represent, transform,solve, apply, prove, communicate’. This happens most readily whenthey are in groups, engage in discussion, make presentations, and inother ways take charge of their own learning.”

€ Mathematics has broad content encompassing many fields.Some aspects of doing mathematics have changed in the last decade.For example, quantitative techniques have permeated almost allintellectual disciplines, and this phenomenon has changed thefundamental mathematical ideas needed. Although traditional topicsremain very important components of the curriculum, there is a shiftin emphasis from a curriculum dominated by memorization ofisolated facts and procedures and by proficiency with paper-and-pencilskills to one that also emphasizes conceptual understanding, multiplerepresentations and connections, mathematical modeling, andproblem solving.

The integration of ideas from algebra and geometry is particularlystrong, with graphical representation playing an important connectingrole. Frequent references to graphing utilities indicate the value ofcomputers with appropriate graphing software and/or graphingcalculators. Topics from statistics, probability, and discretemathematics are now elevated to a more central position for allstudents.

Arithmetic computation is not a direct object of study in the highschool mathematics curriculum; however, conceptual and proceduralunderstandings of number, numeration, and operations and theability to make estimations and approximations to judge thereasonableness of results are strengthened in the context ofapplications and problem solving. Emphasis is placed on the role oftechnology and appropriate concepts and skills related to their use.

€ Changes in technology and the broadening of the areas in whichmathematics is applied have resulted in growth and changes in thediscipline of mathematics itself.

New technology not only has made calculations and graphing easier, ithas changed the very nature of the problems important tomathematics and the methods mathematicians use to investigatethem. Because technology is changing mathematics and its uses,students should learn to use graphing calculators and computers astools for processing information and performing calculations toinvestigate and solve problems.


INTRODUCTION

The visualization approach offered through the use of graphingutilities such as the graphing calculator affords more students greateraccess to more mathematics. With the wide availability of technologycomes additional decision making regarding what skills need to bedeveloped mentally. Some aspects of further development inmathematics are facilitated when students reach an automaticresponse level with certain basic skills.

Meeting the Needs

of All Learners

An important emphasis of this curriculum is the need to deal successfullywith a wide variety of equity and diversity issues. Not only must teachersbe aware of, and adapt instruction to account for, differences in studentreadiness as they begin grade 12 and as they progress, but they must alsoremain aware of the importance of avoiding gender and cultural biases intheir teaching. Ideally, every student should find his/her learningopportunities maximized in the mathematics classroom.

The reality of individual student differences must be recognized asteachers make instructional decisions. While this curriculum guidepresents specific curriculum outcomes for the course, it must beacknowledged that all students will not progress at the same pace and willnot be equally positioned with respect to attaining a given outcome atany given time. The specific curriculum outcomes represent, at best, areasonable framework for helping students to ultimately achieve the key-stage and general curriculum outcomes.

Career

Connections

Mathematics plays a major role in many career options available tostudents. Teachers should take every opportunity to point out tostudents the variety of career options that utilize mathematics in amajor way. All strands of the mathematics curriculum can be directlyconnected with careers. For example, engineering programs require highlevels of algebra, pharmacists, optometrists, carpenters, electricians, andsurveyors use measurement on a day-to-day basis, business draws largelyupon data management, and meteorologists use probability to report onweather data.


CURRICULUM OUTCOMES

Curriculum Outcomes

Curriculum

Outcomes

Framework

The mathematics curriculum is based on a framework of outcomesstatements articulating what students are expected to know, be able todo, and value as a result of their learning experiences in mathematics.This framework comprises statements of the essential graduationlearnings, general curriculum outcomes, key-stage curriculum outcomes,and specific curriculum outcomes. Foundation for the Atlantic CanadaMathematics Curriculum articulates general curriculum outcomes andkey-stage curriculum outcomes. Curriculum guides provide specificcurriculum outcomes for each course, together with elaborations andsuggestions for related instructional and assessment strategies and tasks.

Teachers and administrators are expected to refer to the curriculumoutcomes framework to design learning environments and experiencesthat reflect the needs and interests of the students.

Essential Graduation

Learnings

Essential graduation learnings are statements describing the knowledge,skills, and attitudes expected of all students who graduate from highschool. Essential graduation learnings are cross-curricular in nature andcomprise different areas of learning: aesthetic expression, citizenship,communication, personal development, problem solving, and technologicalcompetence.

Aesthetic Expression Graduates will be able to respond with critical awareness to various forms ofthe arts and be able to express themselves through the arts.

Citizenship Graduates will be able to assess social, cultural, economic, and environmentalinterdependence in a local and global context.

Communication Graduates will be able to use the listening, viewing, speaking, reading, andwriting modes of language(s) and mathematical and scientific concepts andsymbols, to think, learn, and communicate effectively.

Personal Development Graduates will be able to continue to learn and to pursue an active, healthylifestyle.

Problem Solving Graduates will be able to use the strategies and processes needed to solve awide variety of problems, including those requiring language andmathematical and scientific concepts


CURRICULUM OUTCOMES

Technological

Competence

Graduates will be able to use a variety of technologies, demonstrate anunderstanding of technological applications, and apply appropriatetechnologies for solving problems.

Spiritual and Moral

Development

Graduates will demonstrate an understanding and appreciation for the placeof belief systems in shaping the development of moral values and ethicalconduct.

See Foundation for the Atlantic Canada Mathematics Curriculum, pages4–6.

General

Curriculum

Outcomes

General curriculum outcomes are statements that identify what studentsare expected to know and be able to do upon completion of study inmathematics. General curriculum outcomes contribute to the attainmentof the essential graduation learnings and are connected to key-stagecurriculum outcomes. The seven general curriculum outcomes formathematics are organized in terms of four content strands: numberconcepts/number and relationship operations; patterns and relations;shape and space; and data management and probability.

Number Concepts/Number and Relationship Operations

€ Students will demonstrate number sense and apply number theoryconcepts.

€ Students will demonstrate operation sense and apply operationprinciples and procedures in both numeric and algebraic situations.

Patterns and Relations

€ Students will explore, recognize, represent, and apply patterns andrelationships, both informally and formally.

Shape and Space

€ Students will demonstrate an understanding of and apply conceptsand skills associated with measurement.

€ Students will demonstrate spatial sense and apply geometric concepts,properties, and relationships.

Data Management and Probability

€ Students will solve problems involving the collection, display, andanalysis of data.

€ Students will represent and solve problems involving uncertainty.


CURRICULUM OUTCOMES

Key-Stage

Curriculum

Outcomes

Key-Stage Curriculum Outcomes (KSCOs) are statements that identifywhat students are expected to know and be able to do by the end ofgrades 3, 6, 9, and 12 as a result of their cumulative learning experiencesin mathematics. This curriculum guide lists key-stage curriculumoutcomes for the end of grade 12 (see p. 17). Specific curriculumoutcomes are referenced to key-stage curriculum outcomes on these samepages.

Specific

Curriculum

Outcomes

Specific curriculum outcomes are statements identifying what studentsare expected to know and be able to do at a particular grade level, whichcontribute to the achievement of the key-stage curriculum outcomes.

In the table that follows, the Specific Curriculum Outcomes for Course1206 and Course 2206 are listed.


CURRICULUM OUTCOMES

GCO A: Students will demonstrate number sense and apply number

theory concepts.

Elaboration: Number sense includes understanding number meanings, developing multiple relationships amongnumbers, recognizing the relative magnitudes of numbers, knowing the relative effect of operating on numbers,and developing referents for measurement. Number theory concepts include such number principles as laws(e.g., commutative and distributive), factors and primes, and number system characteristics (e.g., density).

The following are the Specific Curriculum Outcomes (SCOs) for Course 1206 and Course 2206.

By the end of Course 2206, students will be expected to

A2 relate sets of numbers to solutions of inequalities

A3 demonstrate an understanding of the applicationsof random numbers to statistical sampling


A2 analyse graphs or charts of situations to identifyspecific information

A3 demonstrate an understanding of the role ofirrational numbers in applications

A4 approximate square roots

A6 apply properties of numbers when operating uponexpressions and equations

A7 demonstrate and apply an understanding ofdiscrete and continuous number systems

A9 explore properties of square roots in applications


CURRICULUM OUTCOMES

GCO B: Students will demonstrate operation sense and apply

operation principles and procedures in both numeric and algebraic

situations.

Elaboration: Operation sense consists of recognizing situations in which a given operation would be useful,building awareness of models and the properties of an operation, seeing relationships among operations, andacquiring insights into the effects of an operation on a pair of numbers. Operation principles and procedureswould include such items as the effect of identity elements, computational strategies, and mental mathematics.

The following are the Specific Curriculum Outcomes (SCOs) for Course 1206 and 2206.


B3 demonstrate an understanding of the relationshipbetween arithmetic operations and operations onequations and inequalities

B4 use the calculator correctly and efficiently

B7 estimate and calculate income and deductions

B8 solve problems involving budgets

B9 analyse situations and make decisions involvingthe financing of purchases

B10 analyse situations and make decisions involvingthe cost of transportation


B1 model (with concrete materials and pictorialrepresentations) and express the relationships betweenarithmetic operations and operations on algebraicexpressions and equations

B3 use concrete materials, pictorial representations,and symbolism to perform operations on polynomials

B5 develop, analyse, and apply procedures for matrixmultiplication

B6 solve network problems using matrices


CURRICULUM OUTCOMES

GCO C: Students will explore, recognize, represent, and apply patterns

and relationships, both informally and formally.

Elaboration: Patterns and relationships run the gamut from number patterns and those made from concretematerials to polynomial and exponential functions. The representation of patterns and relationships will takeon multiple forms, including sequences, tables, graphs, and equations, and these representations will be appliedas appropriate in a wide variety of relevant situations.



C1 express problems in terms of equations and viceversa

C2 model real-world phenomena with linear andquadratic equations

C3 gather data, plot the data using appropriate scales,and demonstrate an understanding of independentand dependent variables and of domain and range

C4 create and analyse scatter plots using appropriatetechnology

C5 sketch graphs from words, tables, and collecteddata

C7 model real-world situations with networks

C8 identify, generalize, and apply patterns

C9 construct and analyse graphs and tables relatingtwo variables

C10 describe real-world relationships depicted bygraphs, tables of values, and written descriptions

C13 determine and interpret the slope and y-interceptof a line from a table of values or a graph

C14 determine the equation of a line using the slopeand y-intercept

C15 develop and apply strategies for solving problems

C16 interpret solutions to equations based on context

C17 solve problems using graphing technology


C6 apply the linear programming process to findoptimal solutions

C8 demonstrate an understanding of real-worldrelationships by translating between graphs, tables,and written descriptions

C11 express and interpret constraints

C18 interpolate and extrapolate to solve problems

C20 solve systems of equations and inequalities, bothwith and without technology


CURRICULUM OUTCOMES



Elaboration: Patterns and relationships run the gamut from number patterns and those made from concretematerials to polynomial and exponential functions. The representation of patterns and relationships will takeon multiple forms, including sequences, tables, graphs, and equations, and these representations will be applied,as appropriate, in a wide variety of relevant situations.



C26 demonstrate an understanding of the differencebetween simple and compound interest

C28 solve simple trigonometric equations


C18 investigate and find the solution to a problem bygraphing two linear equations, with and withouttechnology

C21 explore and apply functional relationshipsinformally

C25 solve equations using graphs

C26 solve quadratic equations by factoring

C28 explore and describe the dynamics of changedepicted in tables and graphs

C29 investigate and make and test conjecturesconcerning the steepness and direction of a line

C31 graph equations and analyse graphs, both withand without graphing technology

C32 plot points, given a situation or a table of values,to help determine if a graph is linear

C33 graph by constructing a table of values by usinggraphing technology and, when appropriate, by theslope y-intercept method

C35 expand and factor polynomial expressions usingperimeter and area models

C36 explore, determine, and apply relationshipsbetween perimeter and area, surface area and volume

C37 represent network problems using matrices andvice versa

C38 transform linear equations into slope y-interceptform

C39 make and test conjectures about how linear andquadratic graphs change as particular values in theequations change

C40 solve linear equations algebraically


CURRICULUM OUTCOMES

GCO D: Students will demonstrate an understanding of and apply

concepts and skills associated with measurement.

Elaboration: Concepts and skills associated with measurement include making direct measurements, usingappropriate measurement units, and using formulas (e.g., surface area, Pythagorean Theorem) and/orprocedures (e.g., proportions) to determine measurements indirectly.



D1 determine and apply formulas for perimeter, area,surface area, and volume

D3 relate the trigonometric functions to the ratios insimilar right triangles

D4 use calculators to find trigonometric values ofangles and angles when trigonometric values areknown

D6 solve problems involving measurement usingbearings and vectors

D7 determine the accuracy and precision of ameasurement

D8 solve problems involving similar triangles andright triangles

D10 determine and apply relationships between theperimeters and areas of similar figures and betweenthe surface areas and volumes of similar solids

D11 explore, discover, and apply properties ofmaximum area and volume

D12 solve problems using the trigonometric ratios

D13 demonstrate an understanding of the concepts ofsurface area and volume

D14 apply the Pythagorean Theorem


D4 solve problems using the sine, cosine, and tangentratios

D5 apply the Law of Sines, the Law of Cosines, and

the formula “area of a triangle 1

sin2

ABC bc A= ” to

solve problems


CURRICULUM OUTCOMES

GCO E: Students will demonstrate spatial sense and apply geometric

concepts, properties, and relationships.

Elaboration: Spatial sense is an intuitive feel for one’s surroundings and the objects in them and ischaracterized by such geometric relationships as (i) the direction, orientation, and perspectives of objects inspace, (ii) the relative shapes and sizes of figures and objects, and (iii) how a change in shape relates to a changein size. Geometric concepts, properties, and relationships are illustrated by such examples as the concept ofarea, the property that a square maximizes area for rectangles of a given perimeter, and the relationships amongangles formed by transversal intersecting parallel lines.



E1 explore properties of, and make and testconjectures about two- and three-dimensional figures

E2 solve problems involving polygons and polyhedra

E6 represent network problems as digraphs

E7 demonstrate an understanding of a proof for thePythagorean Theorem

E8 use inductive reasoning when observing patterns,developing properties, and making conjectures

E10 investigate line and rotational symmetry

E11 draw nets of various polyhedra


E2 represent linear programming problems using theCartesian coordinate system

E3 represent systems of inequalities as feasible regions


CURRICULUM OUTCOMES

GCO F: Students will solve problems involving the collection, display

and analysis of data.

Elaboration: The collection, display, and analysis of data involves (i) attention to sampling procedures andissues, (ii) recording and organizing collected data, (iii) choosing and creating appropriate data displays, (iv)analysing data displays in terms of broad principles (e.g., display bias) and via statistical measures (e.g., mean)and (v) formulating and evaluating statistical arguments.



F1 design and conduct experiments using statisticalmethods and scientific inquiry

F2 demonstrate an understanding of concerns andissues that pertain to the collection of data

F3 construct various displays of data

F4 calculate various statistics using appropriatetechnology, analyse and interpret displays, anddescribe the relationships

F5 analyse statistical summaries, draw conclusions,and communicate results about distributions of data

F6 solve problems by modelling real-worldphenomena

F7 explore non-linear data using power andexponential regression to find a curve of best fit

F8 determine and apply a line of best fit using linearregression with technology

F9 demonstrate an intuitive understanding ofcorrelation

F10 use interpolation, extrapolation, and equations topredict and solve problems

F15 approximate a line of best fit from a scatter plot


F1 draw inferences about a population from a sample

F2 identify bias in data collection, interpretation, andpresentation

F3 demonstrate an understanding of what can beinferred about a population by examining samplemeans and dispersion

F4 demonstrate an understanding of how the size of asample affects the variation in sample results

F5 organize and display information in various ways,with and without technology

F7 draw inferences from graphs and tables

F12 interpret normal curves and standard deviation toexpress levels of confidence

F13 calculate, analyse, and interpret various statistics

F15 design and conduct experiments/surveys toexplore sampling variability

F17 design and conduct experiments/surveys andinterpret and communicate level of confidence


CURRICULUM OUTCOMES

GCO G: Students will represent and solve problems involving

uncertainty

Elaboration: Representing and solving problems involving uncertainty entails (i) determining probabilities byconducting experiments and/or making theoretical calculations, (ii) designing simulations to determineprobabilities in situations that do not lend themselves to direct experiment, and (iii) analysing problemsituations to decide how best to determine probabilities.


By the end of Course 1206, students will be expected to By the end of Course 2206, students will be expected to

G3 graph sample distributions and interpret themusing the language of probability

Independent Study

Students in Course 2206 will also conduct anindependent study.


I1 demonstrate an understanding of a mathematicaltopic through independent research

I2 communicate the results of the independentresearch

I3 demonstrate an understanding of the mathematicaltopics presented by other students


CURRICULUM OUTCOMES

GCO A: Students will demonstrate number sense and apply number

theory concepts.

In the tables that follow, the Specific Curriculum Outcomes for Mathematics 3206 are listed

beside the corresponding KSCOs. Page references for each outcome are included.

Elaboration: Number sense includes understanding number meanings, developing multiple relationships amongnumbers, recognizing the relative magnitudes of numbers, knowing the relative effect of operating on numbers,and developing referents for measurement. Number theory concepts include such number principles as laws(e.g., commutative and distributive), factors and primes, and number system characteristics (e.g., density).

Key-Stage Curriculum Outcomes (KSCO)

By the end of grade 12, students will have achieved theoutcomes for entry–grade 9 and will also be expected to

Specific Curriculum Outcomes (SCO)


KSCO i: demonstrate an understanding of numbermeanings with respect to the real numbers

A1 demonstrate an understanding of and apply zeroand negative exponents (72, 74)

A2 develop, demonstrate an understanding of, andapply properties of exponents (72, 74 )

A3 demonstrate an understanding of the role ofirrational numbers in applications (58)

A6 develop an understanding of factorial notation andapply it to calculating permutations and combinations(110, 112)

A8 demonstrate an understanding of the exponentialgrowth of compound interest (70)

KSCO ii: order real numbers, represent them inmultiple ways (including scientific notation), andapply appropriate representations to solve problems

KSCO iii: demonstrate an understanding of the realnumber system and its subsystems by applying avariety of number theory concepts in relevantsituations

KSCO iv: Some post-secondary intending studentswill be expected to explain and apply relationshipsamong real and complex numbers


CURRICULUM OUTCOMES

GCO B: Students will demonstrate operation sense and apply

operation principles and procedures in both numeric and algebraic

situations.

Elaboration: Operation sense consists of recognizing situations in which a given operation would be useful,building awareness of models and the properties of an operation, seeing relationships among operations, andacquiring insights into the effects of an operation on a pair of numbers. Operation principles and procedureswould include such items as the effect of identity elements, computational strategies, and mental mathematics.



KSCO i: explain how algebraic and arithmeticoperations are related, use them in problem-solvingsituations, and explain and demonstrate the power ofmathematical symbolism



B1 demonstrate an understanding of the relationshipsthat exist between arithmetic operations and theoperations used when solving equations (58, 60)

B3 apply the quadratic formula (56, 58)

B5 demonstrate an understanding of and applycompound interest (70, 76, 78)

B6 determine the amount and present value ofannuities (76, 78)

B7 calculate probabilities to solve problems (102,106, 114)

B8 determine probabilities using permutations andcombinations (114)

B9 perform operations on algebraic expressions andequations (58, 60)

KSCO ii: derive, analyse, and apply computationalprocedures (algorithms) in situations involving allrepresentations of real numbers

KSCO iii: derive, analyse, and apply algebraicprocedures (including those involving algebraicexpressions and matrices) in problem situations

KSCO iv: apply estimation techniques to predict, andjustify the reasonableness of, results in relevantproblem situations involving real numbers

KSCO v: Some post-secondary-intending studentswill be expected to apply operations on complexnumbers to solve problems


CURRICULUM OUTCOMES








C1 model real-world phenomena using quadraticequations (52, 54, 60)

C4 demonstrate an understanding of patterns that arearithmetic, power, and geometric (44, 46, 48)

C5 determine and describe patterns and use them tosolve problems (40, 52, 64, 66, 74)

C6 explore, describe, and apply the Fibonaccisequence (42)

C7 relate arithmetic patterns to linear relations (44)

C8 describe and translate between graphical, tabular,and written representations of quadratic relationships(54)

C11 describe and translate between graphical, tabular,and written representations of exponentialrelationships (64, 66, 72)

C12 describe and apply the characteristics ofquadratic relationships (52, 54)

C13 describe and apply the characteristics ofexponential relationships (64, 66, 72)

C14 determine and interpret x-intercepts of quadraticfunctions (56)

C21 create and analyse scatter plots and determineequations for the curves of best fit, using appropriatetechnology (44, 52, 54, 68)

C23 solve problems involving quadratic equations(56, 58, 60)

C25 solve problems involving exponential equations(68)

C26 solve problems that require the application ofcompound interest (70, 76, 78)

KSCO i: model real-world problems using functions,equations, inequalities, and discrete structures

KSCO ii: represent functional relationships inmultiple ways (e.g., written descriptions, tables,equations, and graphs) and describe connectionsamong these representations

KSCO iii: interpret algebraic equations andinequalities geometrically and geometric relationshipsalgebraically

KSCO iv: solve problems involving relationships,using graphing technology as well as paper-and-penciltechniques


CURRICULUM OUTCOMES

GCO C: Students will explore, recognize, represent and apply patterns





KSCO v: analyse and explain the behaviours,transformations, and general properties of types ofequations and relations

KSCO vi: perform operations on and betweenfunctions

KSCO vii: Some post-secondary intending studentswill be expected to describe and explore the conceptof continuity of a function

KSCO viii: Some post-secondary-intending studentswill be expected to investigate limiting processes byexamining infinite sequences and series

KSCO ix: Some post-secondary-intending studentswill be expected to make connections amongtrigonometric functions, polar coordinates, complexnumbers, and series



C29 analyse tables and graphs to distinguish betweenlinear, quadratic, and exponential relationships (52,64, 66)


CURRICULUM OUTCOMES

GCO D: Students will demonstrate an understanding of and apply

concepts and skills associated with measurement.

Elaboration: Concepts and skills associated with measurement include making direct measurements, usingappropriate measurement units, and using formulas (e.g., surface area, Pythagorean Theorem) and/orprocedures (e.g., proportions) to determine measurements indirectly.



KSCO i: measure quantities indirectly, usingtechniques of algebra, geometry, and trigonometry

KSCO ii: determine measurements in a wide varietyof problem situations and determine specified degreesof precision, accuracy, and error of measurements

KSCO iii: apply measurement formulas andprocedures in a wide variety of contexts

KSCO iv: demonstrate an understanding of themeaning of area under a curve



D2 determine midpoints and the length of linesegments using coordinate geometry (86, 90)


CURRICULUM OUTCOMES

GCO E: Students will demonstrate spatial sense and apply geometric

concepts, properties, and relationships.

Elaboration: Spatial sense is an intuitive feel for one’s surroundings and the objects in them and ischaracterized by such geometric relationships as (i) the direction, orientation, and perspectives of objects inspace, (ii) the relative shapes and sizes of figures and objects, and (iii) how a change in shape relates to a changein size. Geometric concepts, properties, and relationships are illustrated by such examples as the concept ofarea, the property that a square maximizes area for rectangles of a given perimeter, and the relationships amongangles formed by transversal intersecting parallel lines.



KSCO i: extend spatial sense in a variety ofmathematical contexts

KSCO ii: interpret and classify geometric figures,translate between synthetic (Euclidean) andcoordinate representations, and apply geometricproperties and relationships

KSCO iii: analyse and apply Euclideantransformations, including representing and applyingtranslations as vectors

KSCO iv: represent problem situations withgeometric models (including the use of trigonometricratios and coordinate geometry) and apply propertiesof figures

KSCO v: make and test conjectures about, anddeduce properties of and relationships between, two-and three-dimensional figures in multiple contexts

KSCO vi: demonstrate an understanding of theoperation of axiomatic systems and the connectionsamong reasoning, justification, and proof

KSCO vii: some post-secondary-intending studentswill be expected to represent and apply vectors inthree dimensions, algebraically and geometrically



E1 perform geometric constructions and analyse theproperties of the resulting figures (82, 84, 88, 90, 92)

E2 describe and apply symmetry (88)

E5 apply inductive reasoning to make conjectures ingeometric situations (82, 84, 88, 90, 92)

E6 explore, make conjectures about, and apply centresof circles (82, 88)

E7 explore, make conjectures about, and apply chordproperties in circles (82, 90)

E8 explore, make conjectures about, and apply anglerelationships in circles (82, 92)

E10 present informal deductive arguments (80, 82,86, 88, 90, 92, 94)


CURRICULUM OUTCOMES

GCO F: Students will solve problems involving the collection, display

and analysis of data.

Elaboration: The collection, display, and analysis of data involves (i) attention to sampling procedures andissues, (ii) recording and organizing collected data, (iii) choosing and creating appropriate data displays, (iv)analysing data displays in terms of broad principles (e.g., display bias) and via statistical measures (e.g., mean),and (v) formulating and evaluating statistical arguments.



KSCO i: understand sampling issues and their rolewith respect to statistical claims

KSCO ii: extend construction (both manually and viaappropriate technology) of a wide variety of datadisplays

KSCO iii: use curve fitting to determine therelationship between, and make predictions from, setsof data and be aware of bias in the interpretation ofresults

KSCO iv: determine, interpret, and apply asappropriate a wide variety of statistical measures anddistributions

KSCO v: design and conduct relevant statisticalexperiments (e.g., projects with respect to currentissues, career applications, a/o other disciplines) andanalyse and communicate the results using a range ofstatistical arguments

KSCO vi: Some post-secondary intending studentswill be expected to test hypotheses using appropriatestatistics



F2 use curve fitting to determine the equations ofexponential relationships (68)

F3 use curve fitting to determine the equations ofquadratic relationships (54)

F4 interpolate and extrapolate to predict and solveproblems (54, 68)


CURRICULUM OUTCOMES

GCO G: Students will represent and solve problems involving

uncertainty

Elaboration: Representing and solving problems involving uncertainty entails (i) determining probabilities byconducting experiments and/or making theoretical calculations, (ii) designing simulations to determineprobabilities in situations that do not lend themselves to direct experiment, and (iii) analysing problemsituations to decide how best to determine probabilities.



KSCO i: design and conduct experiments and/orsimulations to model and solve a wide variety ofrelevant probability problems, and interpret and judgethe probabilistic arguments of others

KSCO ii: build and apply formal concepts andtechniques of theoretical probability (including theuse of permutations and combinations as countingtechniques)



G1 develop and apply simulations to solve problems(104)

G2 demonstrate an understanding that determiningprobability requires the quantifying of outcomes (98)

G3 demonstrate an understanding of the fundamentalcounting principle, and apply it to calculateprobabilities (98, 100, 106)

G4 apply area diagrams and tree diagrams to interpretand determine probabilities of dependent andindependent events(106)

G6 demonstrate an understanding of the differencebetween probability and odds (102)

G7 distinguish between situations that involvepermutations and combinations (108, 112)

KSCO iii: understand the differences among, andrelative merits of, theoretical, experimental, andsimulation techniques

KSCO iv: relate probability and statistical situations

KSCO v: Some post-secondary-intending studentswill be expected to create and interpret discrete andcontinuous probability distributions and apply themin real-world situations


CURRICULUM OUTCOMES


PROGRAM DESIGN AND COMPONENTS

Program Design and Components

Program

Organization

The mathematics curriculum is designed to make a significantcontribution towards students’ meeting each of the essential graduationlearnings (EGLs), with the communication and problem-solving EGLsrelating particularly well to the unifying ideas of the curriculum (see theOutcomes section of Foundation for the Atlantic Canada MathematicsCurriculum). Specific curriculum outcomes represent the means bywhich students work toward accomplishing the key-stage curriculumoutcomes, the general curriculum outcomes, and ultimately, the essentialgraduation learnings.

Outcomes Framework Examples

Graduates will be able to use the listening,viewing, speaking, reading, and writing modes oflanguage(s) and mathematics and scientificconcepts and symbols to think, learn, andcommunicate effectively.

Graduates will explore, recognize, represent, andapply patterns and relationships, both informallyand formally.

By the end of grade 12, students will be expectedto represent functional relationships in multipleways (e.g., written descriptions, tables, equations,and graphs) and describe connections amongthese representations.

By the end of Course 3206, students will beexpected to describe and translate betweengraphical, tabular, and written representations ofquadratics relationships.

Essential GraduationLearnings (EGLs)—broad cross-curricular

expectations

General CurriculumOutcomes (GCOs)—broad mathematical

expectations

Key-Stage CurriculumOutcomes (KSCOs)—atthe end of grades 3, 6, 9,

and 12

Specific CurriculumOutcomes (SCOs)—for

each grade level

contributes to

contributes to

contributes to



It is important to emphasize that the presentation of the specificcurriculum outcomes in this guide follows a suggested teachingsequence. Student and teacher resources have been developed tocomplement the curriculum guide.

It is recognized that students’ understandings of concepts will vary interms of depth and breadth. Curriculum and Evaluation Standards forSchool Mathematics recommends that the study of mathematics for everystudent revolve “around a core curriculum differentiated by the depthand breadth of the treatment of the topics and by the nature of theapplications” (p. 9). While it is expected that all students will worktoward achievement of the same outcomes, it is recognized that studentswill demonstrate different levels of performance.

Students completing Mathematics 1206 may wish to select Mathematics2206 or Mathematics 3206 since these two courses they can be done ineither order. It is recommended that students do mathematics in eachyear of high school.

All students, at all program levels, will work toward achievement of thesame key-stage outcomes. While the key-stage curriculum outcomes areintended as targets for all students, they will not be expected to achievethem at a single level of performance. As well, there will be an additionalsmall percentage of students who will see their outcomes significantlyaltered in individual educational programs.

Many of the specific curriculum outcomes for each course at the samegrade level are the same, but not all. It is expected that some studentscan move from one program level to another, especially early in the highschool program. This change is typically easier in level I than insubsequent years.

The students in practical-level courses including Mathematics 3206 willbe expected to meet the same key-stage curriculum outcomes and someof the same specific (course) outcomes as those in academic andadvanced level. As well, the instructional environment and philosophyshould be the same at all levels. The significant difference betweenpractical- and academic-level courses will be with respect to the level andnature of performance expected in regard to each outcome.

Advanced

Academic

Mathematics

1204

Mathematics 1206

Mathematics 2205

Mathematics 2204

Mathematics 2206

Mathematics 3205

Mathematics 3204

Mathematics 3206

Mathematics 3207

Program Level Course 1 Course 2 Course 3 Course 4

Mathematics 3103

Practical



By and large, the practical-level courses should be characterized by agreater focus on concrete activities, models, and applications, with lessemphasis given to formalism, symbolism, computational or symbol-manipulating facility, and mathematical structure. Academic- andadvanced-level courses involve greater attention to abstraction and moresophisticated generalizations, while the practical-level course should seeless time spent on complex exercises and connections with advancedmathematical ideas.

Typically, students who choose practical-level courses are those who mayhave experienced considerable difficulty in mathematics throughout theirschooling and may lack confidence in their ability to learn. In addition,their literacy skills may not be on par with students of the same age.They may need more time on new concepts in order to understand themand may need connections presented in more explicit ways. They oftenexhibit lower self-esteem (in relation to mathematics) and require a pacethat accommodates the revisiting and reinforcing of concepts, skills, andknowledge. These students need equal or perhaps greater access totechnology than their peers.

By way of a brief illustration, students at all levels should develop anunderstanding of exponential relationships. Students taking practical-level courses have as much need as others to understand the nature ofexponential relationships, given the place of these relationships inuniversal everyday issues such as provincial and national debt and worldpopulation dynamics. The nature of exponential relationships can bedeveloped through concrete, hands-on experiments and data analysis thatdo not require a lot of formalism or symbol manipulation. The moreformal and symbolic operations on exponential relationships will bemuch more prevalent in the academic- and advanced-level courses.

Content

Organization

The NCTM Curriculum and Evaluation Standards for School Mathematics(1989) establishes mathematical problem solving, communication,reasoning, and connections as central elements of the mathematicscurriculum. Foundation for the Atlantic Canada Mathematics Curriculum(APEF, 1996) further emphasizes these unifying ideas and presents themas being integral to all aspects of the curriculum (see pp. 7–11). Indeed,while the general curriculum outcomes are organized around contentstrands, every opportunity has been taken to infuse the key-stagecurriculum outcomes with one or more of the unifying ideas.

These unifying ideas serve to link the content to methodology. Theymake it clear that mathematics is to be taught in a problem-solvingmode, that classroom activities and student assignments must bestructured to provide opportunities for students to communicate



mathematically, that teacher encouragement and questioning shouldenable students to explain and clarify their mathematical reasoning, andthat the mathematics with which students are involved on any given daymust be connected to other mathematics, other disciplines, and/or theworld around them.

The mathematical content identified in the strands must not be viewedas independent units of study, but must be organized to develop depthas well as breadth. For this depth to be developed, a number of commonconnections must be visible to unify the core content. The unifyingconnections are as follows.

Mathematical

Modeling

Throughout the strands of mathematics that are being studied, studentsneed to see that mathematics is valuable in making predictions in thereal-world. Some basic mathematical structures that are used in modelinginclude graphs, equations, tables, and algorithms. Students need tounderstand also the limitations of modeling real situations, which aremost often very complex.

In some situations the modeling appears to be straightforward. Vectorscan be used to model the movement of an aeroplane in a wind current;exponential functions can be used to model population growth; andquadratics can be used to model trajectory paths. At other times themodel may require transformations in the data. Regression analysisallows us to better understand data from some real situations. Inexamining population growth, exponential functions may produce abetter fit of the data to make predictions. Probability simulation may beused to model processes involving gambling, insurance, and genetics.

Relations and

Functions

An emphasis in the high school mathematics program is the study ofrelationships between two quantities. Across all strands of themathematics program, students need to see the various ways in whichone quantity can vary in relation to another. This study will precede thebasic notion of function, how input and output are related, and howfunctions may be described in various ways, such as verbally, graphically,algebraically, and numerically in tables. A formula such as the one for thearea of a circle, ( ) 2A r rπ= , does not in itself provide a meaning of therelationship. Students need to see how a change in the radius “r” resultsin a corresponding change in the area “A(r)”. This can be describedverbally or graphically.

The type of mathematical reasoning that has students understand thatthe value of one variable may depend on the value of another pervades allstrands of mathematics. In discrete mathematics the relations are notcontinuous, but progress in steps; in exponential functions the functions



are used to model growth and decay. In geometry we can examinerelationships that exist between the image and its object for a giventransformation, and in probability we may also view the probability ofan event as a function of the number of choices available.

Communicating

Mathematics

Communicating in mathematics helps students to develop insight intothe nature of mathematics. Much of mathematics involves solvingproblems where students are required to develop, interpret, and analysealgorithms. When students are given a problem, they should be givenopportunities to share the various ways they solved the problem so thatthey can compare the effectiveness, the efficiency, and the relativeappropriateness of the methods used. It is through this type ofcommunication that students deepen their understanding and extendtheir ability to reason. Technology continues to advance, resulting in achange in the type of problems that we can solve. It is important forstudents to be able to communicate by using technology to solveproblems.

Mathematical arguments help students address questions such as “Howdo I know if I’m correct?” “Is this always true?” “Is there any solution tosatisfy these conditions?” When students are asked to justify a result,they must be able to see how things fit together in a natural way.Mathematical justification communicates a student’s understanding andallows the student to express ideas in many different ways, includingdiscussions of what is and is not accepted. Students may, for example, beasked to clarify when it is appropriate to use an exponential equation.

Mathematical discourse should be part of every lesson, since it promotesboth reasoning and understanding in mathematics. Likewise, whenstudents are asked to write about their mathematical understanding, theyare forced to clarify their thinking in order to reasonably communicateit.

Multiple

Representations

Real understanding in mathematics is present when students are able touse and choose representations to clarify and communicate. Studentswho are in control of their learning may choose or find therepresentation they find most useful. For example, a student who hasstudied the quadratic function demonstrates mathematical power whenhe or she is able to move between the graph and the equation to findsolutions to the quadratic equation or inequality and understands theimplications of these solutions.

An understanding of the multiple ways of representing an idea or solvinga problem as well as the recognition of the equivalence of the variousrepresentations results in deeper understandings of mathematical



structure and process. For example, if students examine the samegeometry concepts from a Euclidean, analytical, and transformationalapproach, they will develop a much stronger intuitive understanding ofthese concepts.

Learning and

Teaching

Mathematics

What students learn is fundamentally connected to how they learn it.The view of learning mathematics as an integrated set of intellectual toolsfor making sense of mathematical situations has created a need for newforms of classroom organization, communication patterns, andinstructional strategies. The teacher is no longer the sole dispenser ofknowledge but is rather a facilitator and educational conductor whosemajor roles include

€ creating a classroom environment to support the teaching andlearning of mathematics

€ setting goals and selecting or creating mathematical tasks to help thestudents reach these goals

€ stimulating and managing classroom discourse so that the studentsare clearer about what is being taught

€ analysing student learning, the mathematical tasks, and theenvironment in order to make ongoing instructional decisions

Good mathematics teaching and learning take place in a range ofsituations. Instructional settings and strategies should create a climatethat reflects the constructive, active view of the learning process. Thismeans that learning does not occur by passive absorption and imitationbut rather as students actively assimilate new information and constructtheir own meanings.

Students’ opportunities to learn mathematics are a function of thesetting and the kinds of tasks and discourse in which they participate.What students learn about particular concepts and procedures and theirown mathematical thinking depends upon the ways in which they engagein mathematical activity in their classrooms. Their dispositions towardmathematics are also shaped by such experiences. Consequently, the goalof developing students’ mathematical power requires careful attention topedagogy as well as to the curriculum.

Mathematics instruction should vary and should include opportunitiesfor group and individual assignments, discussion between teacher andstudents and among students, appropriate project work, practice withmathematical methods, and exposition by the teacher.

Instructional settings should include varied learning environments, whichencourage the development of specific co-operative behaviours. Studentsshould be expected to work together to help each other, and at the sametime they can be expected to complete individual projects. Studentsdevelop strategies and skills in asking questions, listening, showing,explaining, finding out what others think, and determining ways tocomplete a project.



Summary of Changes

in Instructional

Practices

Moving away from

• teacher and text as exclusive sources of knowledge• rote memorization of facts and procedures without understanding• extended periods of individual practising of routine tasks without

applications• instruction based almost completely on teacher exposition• a total emphasis on paper-and-pencil manipulative skill work• testing with the sole purpose of assigning grades

toward practices that include

• the active involvement of students in constructing and applyingmathematical ideas

• problem solving as a means as well as a goal of instruction• effective questioning techniques that promote student interaction• the use of a variety of instructional formats (small groups,

explorations, peer instruction, whole class, project work)• use of computers and calculators as tools for learning and doing

mathematics when appropriate• student communication of mathematical ideas orally and in writing• the establishment and application of the interrelatedness of

mathematical topics• the systematic maintenance of student learnings and embedding

review in the context of new topics and problem situations• assessment of learning as an integral part of instruction

Integrating

Technology

The integration of computers, graphing calculators, video technology,and other technologies into the mathematics classroom allows studentsto

• explore individual or groups of related computations or functionsquickly or easily

• create and explore numeric and geometric situations for the purposeof developing conjectures

• perform simulations of situations that would otherwise be impossibleto examine

• easily link different representations of the same information• model situations mathematically• observe the effects of simple changes in parameters or coefficients• analyse, organize, and display data



All of these situations enhance investigarive learning and problem-solvingpotential. At the same time, teachers have the opportunity to usetechnology to communicate with fellow mathematics teachers, to sharelessons with experts, and to expose their students to information thatwould otherwise be inaccessible.

Students will need to learn to make judgments as to when the use oftechnology is appropriate and when it is not. In all situations, it isimperative that technology be used both as a tool to include, rather thanexclude, students and as a means of creating new teaching strategies, andit must allow students exposure to more mathematical concepts.

Learning

Resources

This curriculum document represents the central resource for the teacherof mathematics with respect to Mathematics 3206 of the high schoolmathematics program. Other resources are ancillary to it. This guideshould serve as the focal point for all daily, unit, and yearly planning, aswell as a reference point to determine the extent to which the curriculumoutcomes have been met.

Nevertheless, other resources will be significant in the mathematicsclassroom. Textual and other print resources will be significant to theextent they support the curriculum goals. Schools, school districts, andDepartments of Education should work together in making professionalresources available to teachers as they seek to broaden their instructionaland mathematical skills. As well, manipulative materials and appropriateaccess to technological resources need to be at hand.

It is highly recommended that teachers familiarize themselves not onlywith Foundation for the Atlantic Canada Mathematics Curriculum, butalso with Curriculum and Evaluation Standards for School Mathematics,Professional Standards for Teaching Mathematics, Assessment Standards forSchool Mathematics, and Principles and Standards for School Mathematics(NCTM, April 2000). Because of the extent of information contained inthese documents, teachers are cautioned that assimilation of the ideascontained will require much reflection, discussion, and rereading. Allhigh school mathematics teachers may wish to join the National Councilof Teachers of Mathematics (NCTM) for professional growth.Membership can include a subscription to The Mathematics Teacher, ajournal that contains a wealth of information and practical teachingsuggestions. Institutional or individual membership can be obtained bytelephoning 1-800-235-7566 (NCTM order office), or by contacting theNCTM representative on your mathematics teacher association’sexecutive.



Assessing and

Evaluating Student

Learning

In recent years there have been calls for change in the practices used toassess and evaluate students’ progress. Many factors have set the demandsfor change in motion, including the following:

€ new goals for mathematics education as outlined in Curriculum andEvaluation Standards for School MathematicsThe curriculum standards provide educators with specific informationabout what students should be able to do in mathematics. These goalsgo far beyond learning a list of mathematical facts; to also emphasizesuch competencies as creative and critical thinking, problem solving,working collaboratively, and the ability to manage one’s own learning.Students are expected to be able to communicate mathematically, tosolve and create problems, to use concepts to solve real-worldapplications, to integrate mathematics across disciplines, and toconnect strands of mathematics. For the most part, assessments usedin the past have not addressed these goals. New approaches toassessment are needed if we are to teach and address the goals set outin Curriculum and Evaluation Standards for School Mathematics.

€ understanding the bonds linking teaching, learning, and assessmentMuch of our understanding of learning has been based on a theorythat viewed learning as the accumulation of discrete skills. Cognitiveviews of learning call for an active, constructive approach in whichlearners gain understanding by building their own knowledge anddeveloping connections between the facts and concepts. Problemsolving and reasoning become the emphases rather than theacquisition of isolated facts. Conventional testing, in which studentsanswered questions for the purpose of determining if they could recallthe type of question and the procedure to be used, provides awindow into only one aspect of what a student has learned.Assessments that require students to solve problems, demonstrateskills, create products, and create portfolios of work reveal moreabout the student’s understanding and reasoning of mathematics. Ifthe goal is to have students develop reasoning and problem-solvingcompetencies, then teaching must reflect such, and in turn,assessment must reflect what is valued in teaching and learning.Feedback from assessment directly affects learning. The developmentof problem-solving and higher-order thinking skills will be a realizedonly if assessment practices are in alignment with these goals.

In planning assessment, it is important to decide whether technologywill be permitted. Certain assessment items become trivial whentechnology is used. It is recommended that when technology is anintegral part of instruction, it should be permitted when those aspectsare assessed. However, there will be times when assessment tools arecreated such that it is inappropriate to permit technology use. Whenthe goal is for students to demonstrate mental facility, calculator usecan interfere.



€ limitations of the present methods used to determine student achievementDoes the present method of assessment provide the student withinformation on how to improve performance? The development ofmethods of assessment that provide accurate information aboutstudents’ academic achievement is much needed. This informationwill guide teachers in decision making to improve both learning andteaching.

What Is Assessment? Assessment allows teachers to communicate to students what activitiesand learning outcomes they truly value. In order for teachers to assessstudents effectively in a mathematics curriculum that emphasizesapplications and problem solving, they need to employ devices thatrecognize the reasoning involved in the process as well as in the product.Assessment Standards for School Mathematics (NCTM, 1995, p. 3) definesassessment as “the process of gathering evidence about a student’sknowledge of, ability to use, and disposition toward, mathematics and ofmaking inferences from that evidence for a variety of purposes.”

Assessment can be informal or formal. Informal assessment occurs whileinstruction is occurring. It is a mind-set, a daily activity that helps theteacher answer the question, “Is what is taught being learned?” Itsprimary purpose is to collect information so that the teacher can makedecisions to improve instructional strategies. For many teachers thestrategy of making annotated comments about a student’s work is part ofthe informal assessment. Assessment must do more than determine ascore for the student. It should do more than portray a level ofperformance. It should direct teachers’ communication and actions.Assessment must anticipate subsequent action.

Formal assessment requires the organization of an assessment event. Inthe past, mathematics teachers may have restricted these events toquizzes, tests, or exams. As the outcomes for mathematics educationbroaden these assessment techniques become more limited. Someeducators would argue that informal assessment provides better-qualityinformation because it is in a context that can be put to immediate use.

Why Should We

Assess Students?

We should assess students in order to

€ improve instruction by identifying successful instructional strategies€ identify and address specific sources of the students’

misunderstandings€ inform the students about their strengths in skills, knowledge, and

learning strategies



€ inform parents of their child’s progress so that they can provide moreeffective support

€ certify the level of achievement for each outcomeIf we assume that assessment is integral to instruction and that it willenable effective intervention in instruction, then it is essential thatteachers develop a repertoire of assessment strategies.

Assessment

Strategies

Some assessment strategies that teachers may employ include thefollowing.

Documenting classroombehaviours

In the past teachers have generally made observations of students’persistence, systematic working, organization, accuracy, conjecturing,modeling, creativity, and ability to communicate ideas, but often failedto document them. Certainly the ability to manage the documentationplayed a major part. Recording information signals to the student thosebehaviours that are truly valued. Teachers should focus on recording onlysignificant events, which are those that represent a typical student’sbehaviour or a situation where the student demonstrates newunderstanding or a lack of understanding. Using a class list, teachers canexpect to record comments on approximately four students per class. Theuse of an annotated class list allows the teacher to recognize wherestudents are having difficulties and to identify students who may bespectators in the classroom. However, for summative purposes, gradesshould reflect the degree to which students achieve the curriculumoutcomes.

Using a portfolio or studentjournal

Having students assemble on a regular basis responses to various types oftasks is part of an effective assessment scheme. Responding to open-ended questions allows students to explore the bounds and the structureof mathematical categories. As an example, students are given a trianglein which they know two sides or an angle and a side and they are askedto find out everything they know about the triangle. This is preferable toasking students to find the side of a triangle in a trigonometry question,because it is less prescriptive and allows students to explore the problemin many different ways, and gives them the opportunity to use manydifferent procedures and skills. Students should be monitoring their ownlearning by being asked to reflect and write about questions such as

€ What is the most interesting thing you learned in mathematics classthis week?

€ What do you find difficult to understand?€ How could the teacher improve mathematics instruction?€ Can you identify how the mathematics we are now studying is

connected to the real world?



In the portfolio or in a journal teachers can observe the development ofthe student’s understanding and progress as a problem solver. Studentsshould be doing problems that require varying lengths of time andrepresent both individual and group effort. What is most important isthat teachers discuss with their peers what items are to be part of ameaningful portfolio and that students also have some input into theassembling of a portfolio.

Projects and investigativereports

Students should have opportunities to do projects at various timesthroughout the year. For example, they may conduct a survey and do astatistical report, they may do a project by reporting on the contributionof a mathematician, or the project may involve building a three-dimensional shape. Students should also be given investigations in whichthey learn new mathematical concepts on their own. Excellent materialscan be obtained from the National Council of Teachers of Mathematics,including the Student Math Notes. (These news bulletins can bedownloaded from the Internet.)

Written tests, quizzes, andexams

Written tests have been accused of being limited to assessing a student’sability to recall and replicate mathematical facts and procedures. Someeducators would argue that asking students to solve contrivedapplications usually within time limits provides us with little knowledgeof the students’ understanding of mathematics. However, a test that isproperly developed can be the most valid and reliable method ofcollecting information about the degree to which students have achievedthe curriculum outcomes.

How might we improve the use of written tests?

€ Our challenge is to improve the nature of the questions being asked,so that we are gaining information about the students’ understandingand comprehension as well as procedural knowledge.

€ Tests must be designed so that questions being asked reflect theexpectations of the outcomes being addressed.

€ Teachers must also reflect on the quality of the test being given tostudents. Are they being asked to evaluate, analyse, and synthesizeinformation, or are they simply being asked to recall isolated factsfrom memory? Teachers should develop a table of specifications whenplanning their tests. A table of specifications will help ensure that thetest surveys all relevant outcomes and that it addresses conceptual,procedural, and problem solving levels.

€ There is a professional obligation to ensure that the assessmentreflects those skills and behaviours that are truly valued. The bottomline is that good assessment is as important as good instruction. Theygo hand in hand in promoting student achievement.


COURSE ORGANIZATION

Course Organization

Course Design This section of the guide presents the Mathematics 3206 curriculumoutcomes that students are expected to achieve during the year. Teachersare encouraged, however, to consider what comes before and whatfollows to better understand how the students’ learnings at a particularcourse level are part of a bigger picture of concept and skill development.

Mathematics 3206 is organized into five units: Patterns, Quadratics,Exponential Growth, Geometry, and Probability. The presentation of thespecific curriculum outcomes in each unit reflects a suggested teachingsequence.

The Two-Page

Spread

The following pages detail curriculum outcomes. Each two-page spreadis dedicated to a small number of specific curriculum outcomes. As muchas possible, connections are made through references to other pages ofrelated outcomes or topics.

At the top of each page the unit topic is presented, with the appropriatespecific curriculum outcome(s) (SCOs) displayed in the left-handcolumn. The second column presents the elaboration of the outcomesand instructional strategies and suggestions, as well as some examplesthat might be used to illustrate achievement of outcomes. Often, theleft-hand pages can be read independently of the right-hand pages. Thethird column includes worthwhile tasks for instruction and/or assessmentpurposes. While the strategies, suggestions, and examples are notintended to be rigidly applied, they will help to further clarify thespecific curriculum outcome(s) and to illustrate ways to work toward theoutcome(s) while maintaining an emphasis on problem solving,communication, reasoning, and connections.

The final column is entitled Suggested Resources and will, with theteacher’s additions, over time become a collection of useful references toresources that are particularly valuable with respect to achieving theoutcome(s) given.


COURSE ORGANIZATION

Unit 1

Patterns

(10 Hours)

PATTERNS


Elaboration—Instructional Strategies/SuggestionsOutcomes

SCO: By the end ofMathematics 3206, studentswill be expected to

Patterns

C5 determine and describepatterns and use them tosolve problems

C5 Mathematics curriculum should be organized around problem solving. Classroomenvironments should be created in which problem solving can flourish and problem-solving strategies are developed and discussed. One such strategy is to search for anddescribe patterns. In the following example, students should make an organized list tohelp them look for the pattern. Once the pattern is discovered, students should use itin solving the problem. For example:

Scientists have invented a time machine (1983). By setting the dial, you can moveforward in time. Set it forward 6 minutes and you will be in the year 1988. Set itforward 16 minutes, and you will be in the year 1993; Set it forward 26 minutes,and you will be in the year 1998; 36 minutes, year 2003. If the machine continuesin this manner, in what year will you be if the timer is set ahead 65 minutes?

This information can be organized as follows:

Students might notice that the numbers (after the second) in the first columnincrease by 10, while the years (after the second) in the second column increaseby 5. They could use this pattern to predict the answer for 66, then adjust for 65.Some students may even plot the ordered pairs and use the graph to verify or makepredictions.

Students should extend the patterns they observe and describe the pattern usingnumbers, words, pictures and/or geometric shapes. For example, see the activities inon the next page.

Time set forward06

162636

65

Year19851988199319982003

?

PATTERNS


Worthwhile Tasks for Instruction and/or Assessment Suggested Resources

Patterns

Performance (C5)

1) For each of the following sets, give the next element of the set. State in your ownwords what you think the patterning rule is.

a) 80, 40, 20, 10, _____Patterning rule: ______________

b) James, Jill, Joan, John, ________Patterning rule: _________

c) 1, 8, 27, 64, 125, ________Patterning rule: ___________

e) 1, 1, 2, 3, 5, 8, 13, ___________Patterning rule: ______________

f) Alvin, Barbara, Carla, Dennis, __________Patterning rule: ________________

g)

Patterning rule: ___________

h) 1 2 3 4

, , , , ...2 3 4 5

Patterning rule: __________i) 6, 30, 150, 750: ___________


j)


2) For parts a) and b), given a few beginning elements and the last element, fill in themissing elements in each pattern.

a)

b)

3) An empty street car picks up five passengers at the first stop, drops off twopassengers at the second stop, picks up five passengers at the third stop, drops offtwo passengers at the fourth stop, and so on. If it continues in this manner, howmany passengers will be on the streetcar after the 16th stop?

Hirsch, Christian R., andRobert A. Laing, eds.Activities for Active Learningand Teaching: Selections fromthe Mathematics Teacher.Reston, VA: NCTM, 1993.

Barry, Maurice, et al,Constructing MathematicsBook 3 - Thomson, NelsonLearning, Chapter 1.

PATTERNS




Patterns

C6 explore, describe, andapply the Fibonacci Sequence

C6 One of the greatest mathematicians of the Middle Ages was Leonardo of Pisa,called Fibonacci. He wrote a book on arithmetic and algebra titled Liber Abacci. Thisbook was influential in introducing to Europe the Hindu-Arabic numerals with whichwe now write numbers. One of the many interesting problems in this book was aboutrabbits. Students might be asked to solve this problem and look for a pattern.

A pair of rabbits one month old are too young to produce more rabbits, butsuppose that in their second month and every month thereafter they produced anew pair. If each new pair of rabbits does the same and none of the rabbits die,how many pairs of rabbits will there be at the beginning of each month?

The number of pairs of rabbits at the beginning of each month form a sequence:1, 1, 2, 3, 5, 8, 13 ... known as the Fibonacci Sequence. Its terms follow asimple pattern. Ask the students to describe this pattern.

The Fibonacci Sequence has shown up in an amazingly wide variety of creations. Forexample:

1) The petals of many flower species—their petals commonly occur only in Fibonaccinumber configurations.

2) The seeds in the flower head of a sunflower spiral in two different directions—thenumbers of spirals are Fibonacci numbers.

3) The same is true in the spiral of pineapples, and pine cones.4) In many trees, the leaves spiral around the stem. The number of turns required to

find a leaf in a position directly above another leaf is a Fibonacci number.5) There is also a connection between the Fibonacci Sequence and musical scales6) Likewise, this sequence is shown in the reproduction of bees.

Students can explore some of the above situations to find out more about how thesequence applies to the situations listed. See the next page for examples.

Students might also explore the ratio between terms in the sequence and connect it to

the Golden Ratio 1 5

2

±

. If the largest square is removed from a rectangle whose

dimensions are in the golden ratio (length: width), another rectangle will remainwhose dimensions are, again, the golden ratio. To explore the ratios betweensuccessive terms, ask students to divide the larger term by the one that precedes it.For example, 5 divided by 3, or 13 divided by 8, or 34 divided by 21. Ask them torecord the answers for several of these ratios as decimals and notice the pattern thatseems to be developing as the terms get larger. Compare the ratios to the decimalapproximation of the golden ratio. (The successive ratios seem to be approaching thevalue of the golden ratio.)

... continued

PATTERNS



Patterns

Performance (C6)

1) In Fibonacci’s rabbit problem,a) How many pairs of rabbits will there be at the beginning of the seventh month?b) How many pairs will there be at the beginning of the 12th month?c) How many pairs of adult rabbits (at least one month old) will there be at the

beginning of the seventh month?d) How many baby rabbits (less than one month old)

will there be?

2) Fibonacci numbers have some remarkable properties.a) Find the missing numbers in this sequence of

sums.b) Describe the pattern.

3 Use the patterns to answer the following questions:a) Guess the sum of the first 10 terms of the Fibonacci Sequence without adding them.b) Write the next line in Pattern B.c) Guess the sum of the squares of the first 10 terms of the Fibonacci Sequence without

adding them.d) Which pattern, A, B, or C, do these figures illustrate:

Pattern A Pattern B Pattern C Pattern D

Dalton, Leroy C., “Algebra inthe Real World”, DaleSeymour Publications, 1983


PATTERNS




Patterns

C4 demonstrate anunderstanding of patterns thatare arithemetic, quadratic,cubic, and geometric

C7 relate arithmetic patternsto linear relations

C21 create and analyse scatterplots and determine equationsfor the curves of best fit,using appropriate technology

C4 Students extend previous knowledge to describe and reason about a variety of contextsusing the mathematical relationships. They should have had experience in creating and usingsymbolic and graphical representation of patterns, especially those tied to linear andquadratic growth. In this course these experiences will be extended to arithmetic, quadratic,and geometric sequences, with particular focus on quadratic and exponential relations.

To begin this unit, students should extend their work with patterns to include investigationof sequences of numbers that fall into two categories:

1) arithmetic sequences (a sequence with a common difference between consecutive terms)2) quadratic sequences (a sequence made up of consecutive terms found by raising

consecutive counting numbers to the same power)

C7 Students should clearly see that an arithmetic sequence leads to a relationship that islinear and can be described as a rule. Students could be given the sequence 2 5 8 11 14and be asked for the 100th term. They can seethat there is a common difference of 3, and bycompleting an organized list they might be ableto predict the 100th term, then the nth term.Teachers should help students to note thatnumbers in an arithmetic sequence have acommon difference, that the common differenceis a constant, and that it is connected to the slopeof the line. When the pattern is expressed with symbols, students need to make sense of theconstant term in the equation and its connection to the pattern. Note: the general formulafor arithmetic sequence is not part of the course.

Students should always be given the opportunity to work with the five representations of aconcept; context, concrete, pictorial, verbal, and symbolic as the concept is being developed.Students might be given the diagram of towers and be asked to construct it with cubes andrecord the heights. Then, they should describe the pattern in words and perhaps attempt tocreate a context for which this pattern exists. They could then graph this relationship.Students could talk about the height of the towers increasing by 3 as the tower numberincreases by 1. They should see that this is connected tothe slope of the graph, and that the slope is thecoefficient of the independent variable in the equation.They could examine the y-intercept and discuss why ithas no meaning in this context. They might concludethat the y-intercept would be –1 and that it wouldrepresent the next height of towers (if that werepossible) in the other direction. Finally, they might beasked to describe the relationship between the tower number in the sequence and the heightof the tower and predict the height of the 10th tower. They might do this by obtaining anequation that would represent the relationship between the number of the towers and theirheights.

C21 As a way of getting their equation, students could use graphing technology. To do this,for example, they could enter the tower number in list 1 and the height of each tower in list2 on a TI-83 and use LinReg to obtain the equation y = 3x – 1. It is important to discusswith students that while LinReg produces an equation, it is meaningful only in the contextfor integers greater than or equal to 1.

... continued

1st term

2nd term

3rd term

5th term

100th term

nth term

(t1)

(t2)

(t3)

(t4)

(t100

)

(tn)

is

is

is

is

is

is

PATTERNS



Patterns

Performance (C4/C21)

1) Complete each sequence and find the nth term.a) 2, 4, 6, 8, 10, _, _, _, ... nth.b) 3, 6, 9, 12, _, _, _, ... nth.c) 2, 7, 12, 17, _, _, _, ... nth.

Performance (C4/C7)

2) Explain why each of the above is called an arithmetic sequence.

Performance (C4/C7/C21)

3) Explore the following dot patterns and determine if they form an arithmeticsequence or not. Explain.

4) If this graph represents a sequence of numbersa) Is the sequence arithmetic?b) What would be the value of the 8th term?c) Describe the sequence in words.d) Describe the nth term.

Performance (C7)

5) a) Create an arithmetic sequence with seven terms, andexplain why it could be described using a linear relation.b) Using y = –3x + 10, develop an arithmetic sequence with six terms.

number of dots on one side: n 1 2 3 4 ... 10... 20

number of dots in array: T

number of dots per side: n 1 2 3 4 ... 10... 20

number of dots in array: s


PATTERNS




Patterns

C4 demonstrate anunderstanding of patterns thatare arithmetic, quadratic,cubic, and geometric

C4 Ask students to examine the dot pattern given below and complete the table ofvalues.

They might notice that each n-value (the number of dots on each side of the square)is squared to get each s-value (the number of dots in the whole array). They could usethis to predict the answers for n = 10 and 20. Students could then describe thepattern in words and arrive at s = n2.

They should be able to describe the differencebetween this pattern and the arithmetic patternslooked at previously. Here, there is no commondifference between successive terms. The differencesare 3, 5, and 7. However, if the common differencesare subtracted students would see a common difference occur at the second level.When this happens students should understand that an equation with an n2

(quadratic) will result. Coming up with the equation is not the goal, however. Theyshould understand that there is a way to describe the relationship with an equation.The sequence where there is a common difference but it is not at the first level is aquadratic sequence.

Students might use cubes to build towers (see the discussion on the previous page) tocompare the growth rate visually between a quadratic relationship and that of anarithmetic sequence (linear relationship).

As suggested above, quadratic sequences have common differences only after the firstlevel of differences. Students should examine the following diagram, extend thediagram, and complete a table. The sequence represents the number of cubes in eachelement of the pattern.

Ask students to determine if this pattern is quadratic (common difference at thesecond level). Ask the students to find a third set of differences in the same way thatthey found the second set of differences. This common difference at the third leveldenotes a cubic relationship. Ask them to describe the pattern in words. Ask studentsto draw a graph of this situation and the rectangular dot pattern at the top of the pageand discuss what is the same about the graphs and what is different. They should nowbe able to contrast the cubic relationship with the quadratic sequence.

n 1 2 3 4 ... 10 ... 20 ... n

s 1 4 9 16

125 216

PATTERNS



Patterns

Performance (C4)

1) The area of the following regions is the number of units inside.

a) Copy and complete the following number sequence for the areas of theseregions. 2 5 __ __ __ __

b) What kind of sequence do the areas form?c) Copy and complete the number sequence of the perimeters 6, 10, _, _, _d) What kind of sequence do the perimeters form? Explain.

2) These figures illustrate a sequence of squares in which the length of the side issuccessively doubled.

a) What are the perimeters of these four squares?b) What happens to the perimeter of a square if the length of its side is doubled?c) What are the areas of these four squares?d) What happens to the area of a square if the length of its side is doubled?

3) The first terms in the sequence of triangular numbers are illustrated by the figuresbelow.

a) Write the five numbers illustrated and continue the sequence to show the nextfive terms.

b) Find the differences between the terms. Is there a common difference?c) Find the difference between the terms in the differences.d) Are triangular numbers terms in an arithmetic sequence? Explain.

4) Teachers could ask students to explain how finding common differences in patternshelps identify the type of sequence, and how that might help them identify whatkind of equation might represent.


PATTERNS




Patterns

C4 demonstrate anunderstanding of patterns thatare arithmetic, quadratic,cubic, and geometric

C4 Students should continue the study of patterns to note that not all sequences ofnumber have common differences. A geometric sequence is a number sequence inwhich each successive term may be found by multiplying by the same number.Students should be able to contrast geometric sequences with other sequences.

The figure at the right shows three stacks of“matho” chips. The stacks illustrate the first threeterms of a geometric sequence. To discover this,students might answer the following questions:1) What are the first three terms?2) Is there a common difference?3) Is there a common ratio?4) What are the next three terms of the sequence?

If the number by which each term is multiplied (common ratio) is greater than 1, thesequence grows at an increasing rate. Students should be asked to describe whathappens to the terms in the sequence if the common ratio is less than 1. They shouldlook at various contexts in which geometric sequences occur.

When a piano is tuned, the first note to be tuned is the A above middle C. It has afrequency of 440 cycles per second. Then the other seven As on the keyboard aretuned so that their frequencies form a geometric sequence. For example, given thepiano keyboard and two of the frequencies:

Students could be asked:

1) What is the common ratio of this sequence?2) Find the frequencies of the other As.3) Write all eight terms of the sequence and then cross out every second term. Do the

remaining terms form a geometric sequence? Explain.4) Sketch a graph of all eight terms. How does it compare to the graphs of arithmetic

and quadratic sequences?

PATTERNS



Patterns

Performance (C4)

1) Teachers could ask students to copy the following geometric sequences, writing inthe missing terms:

2) When a rubber ball is dropped from a height of1280 cm, the heights in cm of the first fewbounces form a geometric sequence:

a) What is the common ratio of this sequence?b) Find the lengths of the third and fourth bounces.

3) Teachers could ask students to tell whether each of the following number sequencesis arithmetic, geometric, quadratic, or cubic. If the sequence is arithmetic, give thecommon difference. If it is geometric, give the common ratio. If other, explainwhy.

a) 5 10 15 20 25

b) 2 8 32 128 512

c) 1 3 6 10 15

d) 80 40 20 10 5

e) 3 20 37 54 71

f) 2 6 24 120 720

g) 60 51 42 33 24

h) 32 48 72 108 162

K

K

K

K

K

K

K

K

4) Teachers could ask students to use graphs to show how there is a difference in anythree patterns above. Explain how the graph helps you decide if the sequence isarithmetic, or geometric.

5) Stella explained that the sequence {7, 43, 125, 271, 499, 827, 1273, 1855 ... } islinear because it has a common difference. Discuss whether you agree with Stella’sreasoning.


PATTERNS




Patterns

Unit 2

Quadratics

(15-20 Hours)

QUADRATICS




Quadratics

C29 analyse tables and graphsto distinguish between linear,quadratic, and exponentialrelationships

C12 describe and apply thecharacteristics of quadraticrelationships

C1 model real-worldphenomena using quadraticequations

C21 create and analyse scatterplots and determineequations for the curves ofbest fit using appropriatetechnology


C29/C12/C1 In Mathematics 1206 and in the preceding unit of this course, students haveanalysed and applied arithmetic sequences and have connected them to linear relations,reaffirming their understanding that a linear relation represents a constant growth rate. Inthis course, as students begin to study quadratic relationships, they should note connectionsto these quadratic sequences (common difference at the second level), examined in theprevious chapter.

C29/C12/C1/C21 In this unit students should examine situations presented in graphs andtables, and determine if they can be described as a linear or quadratic relationships. Studentsshould be able to determine equations for patterns using regression and whether therelationship has a maximum or minimum value by examining the numerical coefficient ofthe x2 term. If the numerical coefficient of x2 is negative, then the graph is reflected in the x-axis causing the vertex to be at the highest point of the graph, giving a maximum value. Theyshould also be able to get this information from a table. Finally, students should be able tosolve problems by interpolating or extrapolating using the graph or equation.

C29/C12/C1/C5 The campsite problem (on the next page) asks campers to stake out theircampsite with 50 metres of string with which they are to create the rectangular boundary.One side does not require string, being a river bank.Students must find the length and widthmeasurements to maximize the area of their campsite.As students plot a graph of width versus area they willnote that as they increase the width a unit at a time,the area of the campsite does not increase at a constantrate. They will see as they continue to plot orderedpairs that the graph will take on a parabolic shape.Since the graph has its vertex at the highest point,students should expect the coefficient of x2 in theequation to be negative.

C5/E2 From the table of width versus area,students should notice the symmetry in thearea values. They should also notice that asthe width increases metre by metre the areaincreases at a different rate each time.

They should be able to say that the relationship isquadratic and that the graphical representationwould be parabolic.

A similar pattern can be seen in the table at theright. Students should be able to predict the next y-value using common differences between successivey-terms, the next two y-values will be –12 and –10.

Students should focus on the visual patterns and the symmetry in both the graph and thetable. This is particularly evident in the table and graph shown below.

QUADRATICS



Quadratics

Pencil and Paper (C29/C12)

1) Which of the following tables do you think will produce a i) linear relationship? ii)quadratic relationship? iii) neither? Explain.

2) Compare the data in these tables. Which table(s) do you think represents aquadratic relationship? Explain your thinking and tell whether the coefficient ofthe quadratic term is positive or negative.

a)

b)

c)

a)x 0 1 2 3 4 5y 2 4 6 8 10 12

x –5 –3 –1 1 3 5y 16 12 8 4 0 –4

b) c)x –3 –2 –1 0 1 2y 18 8 2 0 2 8

Pencil and Paper (C1/C29/C12)

3) The following table represents the height ofthe water in a tank as the tank is beingdrained. Do you think the pattern in thegraph represents a quadratic relationship?Explain.

4) Create a graph and/or a table of values that represents a relationship that is linear,quadratic, or neither. Give it to your partner and ask him/her to determine which itis and explain why.

Pencil and Paper (C21)

5) For questions 1, 2, and 3 above, determine the equation for the curve of best fitusing technology.

Performance (C1/C5/E2)

6) As campers arrive at By the River campsite, they are given string (50 m) and fourstakes with which they are to mark out a rectangular region for their tent. Antoinesuggests that they use the river for one boundary, which would give more string forthe other sides.a) Andrée wants to make the width (the sides perpendicular to the river) 10 m.

What will be the length of the other side?b) Describe fully in words how the length of the boundary changes as the width

increases through all possible values.c) Find the area enclosed by the boundary for each different width.d) Sketch a graph to show how the area enclosed changes as the width increases.e) Crystal wants to find the dimensions that produce the greatest area.

i) Describe in words a method by which you could find this length and width.ii) Use your method to help Crystal.

x 1 2 3 4 5 6 7

y 90 75 60 45 30 15 0

x 0 10 20 30 40 50

y 5 11 16 20 23 25

y 0 10 18 24 28 30 30 28 24 18 10

x 0 1 2 3 4 5 6 7 8 9 10

Time

Height(min)

(cm)

0 1 2 3 4 5 12 7.6 4.2 1.8 0.4 0


QUADRATICS




Quadratics

C21 create and analyse scatterplots and determine equationsfor the curves of best fit usingappropriate technology

F3 use curve fitting todetermine the equations ofquadratic relationships

F4 interpolate andextrapolate to predict andsolve problems

C12 describe and apply thecharacteristics of quadraticrelationships


C8 describe and translatebetween graphical, tabular,and written representations ofquadratic relationships

C1/C21/F3 To get the equation for a parabola, students should create a scatter plot fromdata. The focus should be on using the given information to generate enough data points sothat students can determine the curve of best fit for the scatter plot using appropriatetechnology. They will use “quadratic regression” to determine the equation, unless thepattern is obvious.

F3/F4 Students would choose quadratic regression when they are quite sure of a parabolicshape.

The emphasis should be on exploring the visual display of the relationship. Students shouldbe aware that this relationship can be represented by an equation they could generate usingtechnology. They should use the graph and the equation to make predictions and answerquestions. Students should have a variety of experiences exploring the use of quadraticregression. Students should be given contexts where it is of interest to them to interpolateand extrapolate.

Example:

A company is planning to make 15-cm circular personal pizzas and party pizzas (50 cm).They want to determine what price to charge for each new size pizza. Their current pricelist for other circular pizzas is given in the table below:

Ask students to create a graph of the relationship and estimate the price of 15-cm and50-cm pizzas using the graph. Have them predict the price of a 40-cm pizza. Studentsshould note that it makes sense that this relationship is quadratic, since the area of a circleis obtained by squaring the radius, and squaring a variable leads to a quadraticrelationship. An extension of this activity might be to provide various brochures fromlocal pizza places and have different groups of students find the mathematical model foreach place. They should present their findings as an advertisement for a pizza place that isintroducing their new personal and party-size pizzas.

C12 Students should, through a variety of experiences with relations, come to recognize theelements in a real-world problem that suggest a particular model, e.g., area suggests aquadratic function, since it changes at a different rate as the width of the rectangle increases.Trajectory suggests a quadratic function just in its natural going-up-and-down pattern, witha maximum value, denoted in the equation by a negative coefficient of x2.

C1/C8 Students must be able to model situations with and solve problems using a quadraticrelation. Situations may be presented in words (or words and equations) or by graphs and/ortables of values. When solving a problem, students might be expected to use an equation topredict or to get a table to see the maximum value. They might begin by collecting data orreading the given data, or by creating graphs using appropriate scales, domains, and ranges.For example, the problem might be to calculate the “hangtime” of a punt in a football game.Students can picture the ball after it is kicked. The path it follows “looks” parabolic. Aproper domain can easily be selected since the ball is not likely to hang longer than 6seconds. The range is limited by the height the ball will reach.

Diameter (cm) 25 30 38 46Price in dollars 5.25 6.33 9.00 12.93

QUADRATICS



Quadratics

Performance (C1/C21/F3/F4/C8)

1) Teachers could ask students to collect data, sketch a graph, and model the followingsituation in order to predict answers and solve the problem:Extend a 3 m wire from the back of a desk to the top of the chalkboard and mark10 cm intervals. For a starting position, let a carrier roll along the wire, and timehow long it takes for the carrier to reach each of the marks along the wire up to 2m. From this data, predict how long it will take to reach 2.5 m and 3 m. Verifyyour prediction.

2) Chantal pulled the plug in her bathtub and watched closely as the water drained.She made marks on her tub and used them later to determine the quantity ofwater remaining in the tub at various time intervals. The table contains the datashe determined.

a) Teachers could ask the students to model the data with an equation usingtechnology expressing litres of water remaining in terms of time.

b) Teachers could ask the students how much water was in the tub when the plugwas pulled how long will it take to empty, and to explain why they chose tomodel the data with the function they chose.


3) This picture represents the path of a ball as it flies through the air.a) Ask students to describe how the height of

the golf ball changes.b) Ask them to sketch a graph to illustrate their

description and explain why they drew it likethey did.

c) When a golf ball travels through the air (goesway up into the sky, then comes back to landon the ground) do you think it maintains thesame speed at all times? Explain.

d) Where in its flight is the speed of the ball theslowest? Explain.

4) I threw a ball up into the air from the roof of my house. It landed on the street.Sketch a picture of the flight of the ball. How is it the same as the flight of the golfball in question #3? How is it different?

5) From the table below, describe in words the relationship between an embryo’slength and age.

Time in seconds 15 25 48 60 71 100 120 130 150 180 190Water (litres) remaining 55 51.1 42.6 38.6 35 26.5 21.4 18.8 14.6 9.5 7.9

Age (months) 2 3 4 5 6Length (cm) 1 9 14 18 20


Computer Software:

1. Parris, Richard, Winstats(available free from http://math.exeter.educ/rparris)– This is excellent for

scatterplots andregressions

2. Parris, Richard, Winplot(available free from http://math.exeter.edu/rparris– This is excellent

graphing software

QUADRATICS




Quadratics

C14 determine and interpretx-intercepts of quadraticfunctions

B3 apply the quadraticformula

C23 solve problems involvingquadratic equations

C14 In a previous course, students explored the factors of a quadratic equation and theirconnection to the horizontal axis intercepts on graphs. In this course students will beexpected to solve all quadratic equations by using graphs (e.g., reading horizontal axisintercepts) or by using the quadratic formula.

Students should explore horizontal axis intercepts in a meaningful context. For example, aproblem involving diving off a cliff gives students an opportunity to explore the vertexcoordinates and horizontal axis intercepts in a meaningful way. Because it is a graph ofheight versus horizontal distance, this graph is in fact a picture of the event.

The vertex coordinates represent the maximumheight of the diver, and the horizontal distancetravelled to obtain the maximum height. Thehorizontal axis intercepts represent the horizontaldistance travelled to the water-entry point.

The equation h = 88 + d – 0.85 d2 expresses thedistance above the water h versus d, the horizontal

distance travelled. Students should graph the relationship and explore the values at the h-intercept, the top of the curve, and the d-intercept. Interpret each of these values withrespect to the problem. A lot of discussion and guidance will be required initially to assiststudents in determining an appropriate window setting for their graphing calculator.

B3/C23 In the above example, most students would use technology to enter the equation,draw its graph, and trace to get values for the vertex and the d- and h-intercepts. In aprevious course students connected horizontal axis intercepts to solve equations where h = 0.To motivate the need for the quadratic formula to be introduced, students should be given aquadratic equation to solve for which technology can produce only an approximate answer.To get the exact answer requires an algebraic process. Give the students the quadraticformula (see below) and ask them to use it to obtain values for the independent variable,then compare these unknowns to the values approximated from the graph. In previouscourses, students also learned to find the horizontal axis intercepts (x-intercept) usingfactoring. Students should understand that the quadratic formula can be used instead offactoring, especially, when factoring is not possible. When ax2 + bx + c = 0 and 0a ≠ is

solved for x, the result is 2 4

2

b b aca

− ± − .

It is only necessary for students to be able to use and apply the quadratic formula. Studentsmight then be asked to use the quadratic formula to determine where the diver enters thewater. Since the quadratic formula has a ± symbol, two possible roots will result. Only onewould represent where the diver enters the water. The other is an inadmissable root. Takethis opportunity to talk about inadmissable root and how, in real context, often the use ofonly one of the roots of the equation is appropriate. Students need to use the formula severaltimes to get used to substituting the values for a, b, and c. As well, they should check theanswers with the graphing calculator to ensure that the use of the formula is correct and tovalidate for themselves that the formula really does work.

... continued

QUADRATICS



Quadratics


1) Richard and Elaine shot two missiles at a target whose coordinates are (71, 0).Richard’s missile followed a path defined by the equation h = –t2 + 60t + 828,while Elaine’s missile followed a path defined by the equation h = –t2 + 60t + 892,where h is height in metres and t is time in seconds. Both missiles overshot thetarget. Which missile overshot the target the most? Richard said that his missilewas three times further from the target than Elaine’s. Is he correct? Justify.

Performance (C14/B3)

2) a) Solve the following quadratic equations, then use the solution(s) to match eachequation with its corresponding graph, if possible:i) x2 – 3x – 10 = 0ii) x2 = 2x +15iii) x2 – 25 = 0iv) x2 + x = 12v) x2 + x + 12 = 0vi) x2 = 3x + 5


3) In 1919, Babe Ruth hit a very long home run in a baseball game between theBoston Red Sox and the New York Giants. The trajectory of the ball is given bythe equation , where x represents the horizontal distance(in feet) and y the vertical distance (in feet) of the ball from home plate.a) What was the greatest height reached by the ball? [150]b) How far from home plate did the ball land? [591]c) At what height was the ball when it crossed the plate? [3]

... continued


2 0.0017 + + 3y x x= −

QUADRATICS




Quadratics

B3 apply the quadraticformula


A3 demonstrate anunderstanding of the role ofirrational numbers inapplications

B1 demonstrate anunderstanding of therelationships that existbetween arithmeticoperations and the operationsused when solving equations

B9 perform operations onalgebraic expressions andequations

A3 Since irrational numbers arise when solving quadratic equations, discussion shouldcentre around whether an exact or an approximate solution is appropriate. Students willalways be expected to give their answers using significant digits correctly. Students should befully cognizant of the inaccuracies caused by rounding errors.

B1/B9 Consider this example:

2

2

2

if 25

5, and discuss the reasoning that should occur:

if 5, then 25

if 5, then 25

x

x

x x

x x

== ±

= =

= − =

This brief activity provides opportunity for worthwhile discussion that solidifiesstudents’ understanding of operation sense and the underlying importance ofdefinitions and “order of operations.” For example, the difference between –22 and (–2)2 can be understood by recognizing what the base is and then performing thesquaring operation first.

Students should also understand that if (x + 2) (x + 3) = 0, then one or both of thefactors (x + 2) and (x + 3) must be zero, and how this leads to the solution that hastwo possible answers. For example:

{ }

if 2 0

2

if 3 0

3

2, 3

x

x

x

x

+ == −

+ == −∴ − −

QUADRATICS



Quadratics

Pencil/Paper (B3/A3)

4) Hector has been told that the width (x) of arectangular field can be found using the equation3x = 5 – 2x2. On the right, he is using the quadraticformula to find the width.a) State the equation.b) Explain what Hector did to get the equation in

step 1.c) Explain what Hector did to get the equation in step 2. Is he correct? Explain.d) Ask students to solve the original equation. Ask them what number system is

represented in the solution?

5) Revisit question 1) on the last two-page spread. Ask students to use the quadraticformula to determine the time it would take for each missile to hit the ground.Compare the answers obtained this time to the answers obtained the first time.Compare your answers. How are they the same? Different?

6) a) Explain how the graphing calculator displays the x-intercepts, and how this isdifferent than the answer you get using the quadratic formula.

b) Redo question 2) on the last two-page spread using the quadratic formula.Compare your answers.


2

2

3 5 2

step 1: 2 3 5 0

3 9 40step 2:

4

x x

x x

= −

+ − =

− ± +

QUADRATICS




Quadratics



B9 perform operations onalgebraic expressions andequations

B1 demonstrate anunderstanding of therelationships that existbetween arithmeticoperations and the operationsused when solving equations

C1/C23 Students should be involved in solving a variety of problems using varioustechniques that involve quadratic equations. The expectation is that students would often usetechnology as a tool in solving the problems. If an equation is required, students shoulddetermine the equation using quadratic regression. To solve the equation, they can let y = 0and then use the x-intercepts of the graph, or the quadratic formula, to obtain the roots.Spreadsheets might be a tool that could help students solve problems.

The “campsite” problem discussed on pp. 52-53, and “pasture problem” problem discussedon the opposite page provide the opportunity for students to use a trial and error approach,and perhaps a spreadsheet to help with the calculations. In the campsite problem, theymight begin by trying a “width” dimension of 5 m. This should lead them to reason that thelength would have to be 50 m less twice the 5 m or 40 m. Students might organize the trialsinto a table with headings Width, Length, and Area. Some students may wish to go furtherand set up a spreadsheet, or use the “table” feature on the graphing calculator. Forinstructions on how to use the T.I-83 plus graphing calculator for a similar activity, refer tothe handout from the Newfoundland and Labrador Department of Education’sMathematics 3204/3205: Supplementary Support Materials, Unit I, Investigation 4. Thissupplementary document was sent out to all schools and can also be found on theDepartment of Education website. This approach would require them to establish formulasfor the three categories “x” for width, “50 – 2x” for length, and “x (50 – 2x)”, for area. Thenas they enter different width values, the length and area would automatically appear on thespreadsheet.

Others may graph the width versus the area and find the maximum by tracing the path tothe curve until they find the highest point. They would then be expected to interpret it asthe greatest area.

C23/B9/B1 When using a quadratic equation to solve a problem students might first haveto rearrange the equation into general form (ax2 + bx + c = 0), so they can determine thevalues for a, b, and c. They should be aware of the equation-solving process that allows themto manipulate the equation into the appropriate form. For example, if given the equation , students might first expand the left hand side, thus enabling them torearrange the equation so that the sum of the terms will be zero.

C1/C21/C23/F3 A trajectory problem usually includes an equation that represents thepath of the object. Students might evaluate this equation for various heights. The graph ofeach equation can be traced to find solutions (x-intercepts) of the corresponding quadraticequation or maximum heights. Other trajectory problems may give certain information andexpect students to determine the equation that describes the path. This can be done byentering the data points in lists, creating a scatter plot, and finding the curve of best fit usingquadratic regression. Sometimes students are asked to find the maximum height. Knowingthe time it takes to reach a maximum height is halfway between the x-intercepts (symmetryof the parabola) might provide another way for students to calculate the maximum height(assuming the x-intercepts exist).

( ) 22 5 3x x− + =

( ) 2

2

2

2 5 3

2 10 3

0 3 2 10

x x

x x

x x

− + =

− − =

= + +

QUADRATICS



Quadratics

Performance (C1/C23/B9/B1)

1) A football is kicked into the air and follows the path h = –2x2 + 16x, where x is thetime in seconds and h is the height in metres.a) What is the maximum height of the football?b) How long does the ball stay in the air?c) How high is the ball at 6 seconds?d) How long does it take the ball to reach a height of 15 m?

2) Farmer Brown has many hectares of pasture that have notbeen fenced. His sister Ethel asks her brother if she can usesome of his land to keep her cattle. He decides that he canspare some pasture area down by the old stone wall. He has440 m of fencing. Ethel wants to use the 440 m of fence to create three walls (thestone wall will be the fourth) of a rectangular area.a) Create a table with the following headings: width, length, area.b) Create a graph of area versus width.c) Find the measurements of this rectangle so that she will have a maximum

amount of grass.d) What will be the area? How do you know it’s a maximum? Explain.

3) Tracie and Nathaniel are doubling the floor area of their camp, which now measures48 square metres. The equation x2 – 8x – 16 = 96 represents the new floor(enlarged area) whose width is x metres. Use the equation to find the dimensions ofthe new floor.

Enrichment (C1/C23/B9/B1/C21/F3)

1) An ice cream specialty shop currently sells 240 ice cream cones per day at a priceof $3.50 each. Based on results from a survey, for each $0.25 decrease in pricesales will increase 60 cones per day. If the shop pays $2.00 for each ice cream cone,what price will maximize the profit?


QUADRATICS




Quadratics

Unit 3

Exponential Growth

(20 Hours)

EXPONENTIAL GROWTH




Exponential Growth


C13 describe and apply thecharacteristics of exponentialrelationships

C11 describe and translatebetween graphical, tabular,and written representations ofexponential relationships


C29/C13/C11/C5 Having already studied linear and quadratic relations in this andprevious courses, students should extend their study of relationships to those that areexponential. Students have connected linear functions with arithmetic sequences andquadratic functions with quadratic sequences, and now they should connect exponentialfunctions with geometric sequences. For example, they might want to revisit the paperfolding activity (from an earlier course) that produced data for the thickness of a simple sheetof paper. When they graph this relationship, they may not be able to distinguish it from aquadratic relationship at first. The students fold the paper in half, in half again, etc. Theycan easily see from the table that after folding the paper in half the fifth time, they have athickness 32 times what they started with.

Ask them to extend the table using their understanding of the pattern. Have students createand graph a similar table for the quadratic relationship y = x2. Ask them to describe whathappens to both graphs and tables after the seventh value. When asked to find a pattern inthe data, students will try constant growth and see that the data is not linear. They might trycommon difference and find that the data is not quadratic. Students should note thecommon ratio, e.g., and hence connect this kind of relationship to geometric sequence.

C13 Teachers should talk to students about the growth characteristic of exponentialrelationships. As one variable increases at a constant rate, the other increases or decreases asa multiple of the previous one, e.g., the common ratio between terms for the situation aboveis 2.

C29/C13/C11/C5

Return to the paper-folding activity and focuson the area of the paper. Students shouldnotice that as the paper is folded in half, the

area gets smaller by a factor of 12

each time

(e.g., 12

the original, 14

the original, 18

the original, ...). They

can compare this situation, its table, and graph with thethickness situation and note that both are exponential—thefirst, a growth; the second, a decay. Students could be asked toconsider if the area would ever become zero.

This might help them understand that an exponentialrelationship should always approach an asymptote (a line towhich the graph approaches at an extreme value but nevertouches).

Number of folds 0 1 2 3 4 5Thickness 1 2 4 8 16 32Powers of 2 20 21 22 23 24 25

Number of folds 0 1 2 3

Area 1 12 1

4 1

8

Powers of 2 20 2–1 2–2 2–3

EXPONENTIAL GROWTH



Exponential Growth

Paper and Pencil (C29/C13)

1) Given these three tables of data taken from three different experiments,a) which, if any, do you think represent an exponential relationship? Explain your

reasoning. [Ans.: ii]b) if any are exponential, explain whether they are growth or decay. [ans.: (ii) is

growth]

(i) (ii)

(iii)

2) Which of these graphs might represent an exponential relationship? Explain yourreasoning.

Paper and Pencil (C11/C5)

3) Take a sheet of paper and fold it in half. You now have two sections. Fold in halfagain. You now have four sections, and so on. Complete the table

a) Explain how to express all the S-values as an exponential relationship with a baseof 2.

b) Use what you found in (a) to predict the number of sections after 8 folds; 10folds. Discuss whether this many folds would be possible.

c) Sketch a graph using the values you have obtained so far. Should you join thedots? Explain.

d) What kind of relationship does your graph represent—linear, quadratic, orexponential? Explain how you know. Is it growth or decay? Explain.

Paper and Pencil (C29/C13/C11/C5)

4) Ima Clever is the hottest math teacher around. One major school is so anxious tohire her that they offered her a choice of three salary options.

Option 1: $30 for the first day of work, but overall earnings double for eachadditional day of workOption 2: Three cents for the first day of work, but overall earnings triple for eachadditional day of workOption 3: A flat rate of $300 000 a day for the 195-day school year.

Ask students to determine which contract Ima should sign. Have them explain fullyusing graphs, tables, and written reports.

Number of folds: n 0 1 3 4 5 6Number of sections: S 1 2 4 8

x 1 3 5 7 13 –11

y –4 –32 –84 –160 –532 –340

x 1 3 5 7

y 13 117 1053 13689

x 1 3 5 7 13 –11

y –17.5 .825 1.825 2.825 –5.825 –6.175


Computer Software:

1. Parris, Richard, Winstats(available free from http://math.exeter.educ/rparris)– This is excellent for

scatterplots andregressions

2. Parris, Richard, Winplot(available free from http://math.exeter.edu/rparris)– This is excellent

graphing software

(i) (ii) (iii) (iv)

EXPONENTIAL GROWTH




Exponential Growth



C5 determine and describepatterns and use them to solveproblems


C29/C13/C5 Having just studied quadratic relationships in the previous unit,students should compare the two functions (quadratic and exponential) withparticular focus on their growth characteristics. For example, consider the allowanceproblem:

Byron and Jethro were comparing their allowances. Byron receives 1 cent on thefirst day, 4 cents on the second, 9 cents on the third, 16 cents on the fourth day,and so on—each day receiving an amount equal to the the square of the day of themonth in cents. Jethro convinced his parents to pay him 1 cent on the first day ofthe month, 2 cents on the second day, 4 cents on the third day, and so on—eachnext day receiving double the amount than on the previous day. Who has moremoney accumulated at the end of one week, two weeks, three weeks, a month?

C11 Students might want to model the growth of the allowances using towers ofcubes and create tables to compare the accumulated amounts of money, for bothByron and Jethro as the month progresses. Students should try to describe the rates ofgrowth in words and draw graphs.

Accumulated Amounts

day 1 2 3 4 5

Byron 1 5 14 30 55

Jethro 1 3 7 15 31

C29/C13/C5/C11 Simple and compound interest situations provide good contextsfor examining and comparing rates of growth that are linear and exponential in nature.From a practical situation of $500 growing at 6% compounded annually, a graph canbe generated. Students could consider the value of money after three years andcompare it to the value in a simple interest situation. A graphical display should beexamined.

Students should feel comfortable moving from one representation to the other (graph,table, context, verbal, symbol) regardless of what representation of the exponentialrelationship they are given. Given the graph, they should be able to recognize its shapeand represent it with a table or in words. Given the table of values, they should beable to see the exponential pattern and describe it in words.

Simple Interest Compound Interest

EXPONENTIAL GROWTH



Exponential Growth


1) A culture of bacteria was grown in a laboratory. The table below shows the number ofbacteria present at different times.

a) Begin by studying the table. Can you see any pattern in the data?b) Compare the numbers of bacteria in the culture at zero hours and at three hours. In

the same way, compare the numbers at one and four hours, two and five hours, threeand six hours, four and seven hours. What do you notice? Use the pattern you havefound to predict the number of bacteria after 8, 9, 10, and 11 hours.

c) What is the approximate growth ratio (population increase per hour)? What do younotice? Use this to predict the number of bacteria after 8, 9, 10, and 11 hours. Doyour answers agree with your earlier predictions?


2) Why do you think that the Canadian government declared chain letters illegal? Usewords, graphs, tables, and/or equations to help explain your answer.

A chain letter usually contains a message that encourages the reader to send money ora gift to the top name on a list of up to 10 names, then to erase the top name and addhis or her own name to the bottom of the list. Then the reader is to send this letter to20 friends, asking each of them to do the same.

Performance (C29/C13/C5/C11)

3) Match the situation given with the relationship (linear, quadratic, exponential) thatwould best describe the value of the investment rate.

Situation 1: Billy invests in his friend’s new cyclo-motor machine. Billy gives $500 tohis friend. His friend says that each month he would set aside 1% of Billy’sinvestment and return the $500 plus the amount set aside at the end of two years.Situation 2: Maria invests in Sally’s ITS business. Sally says she will repay Maria theamount of money invested plus the amount earned at 12% yearly interest. The firstmonth she will calculate 1% of the amount invested and add it to the invested amount sothat the next month the 1% will be calculated on the ‘new’ total, thus earning interest onthe amount plus interest.Situation 3: Sally offers Harold a different deal. She says she will repay him with themoney that accumulates after she puts $1 into an account and each month triples theamount that was put in the last time.

a) Which way would you prefer to invest your money? Explain your reasoning. Includetables of values and graphs.

4) Given the grapha) Prepare a table of values.b) Explain how you know that the relationship in the

graph is exponential.c) Describe the relationship using the context of

compound interest.

Time (hours) 0 1 2 3 4 5 6 7

Number 100 126 160 200 250 316 400 500


EXPONENTIAL GROWTH




Exponential Growth

C21 create and analyse scatterplots and determineequations for the curves ofbest fit using appropriatetechnology

F2 use curve fitting todetermine the equations ofexponential relationships

F4 interpolate and extrapolateto predict and solve problems

C25 solve problems involvingexponential equations

C21 Exponential relationships, compound interest, population growth, and theallowance problem provide interesting contexts. If appropriate technology is available,students can conduct an experiment and collect data using technology. Suchexperiments should result in data that looks exponential (bacteria growth orradioactive decay; heat build-up in a car on a sunny day; or the cooling of a cup ofcoffee as it sits untouched). Again the focus should be on identifying independent anddependent variables and recognizing that the relation has a pattern. Furthermore, thepattern can be described by a curve on a graph and by an equation. If the contextsuggests that the pattern approaches an asymptote, then students should chooseexponential regression. All students should be able to generalize the exponentialpattern to a function using graphing technology.

F2/F4C25 From the graphs, students should be able tointerpolate and extrapolate answers to help them seevalues that indicate different growth rates. They shouldsketch and use technology to obtain the curve of bestfit. For example, students might conduct anexperiment to test how the horizontal distance thata ball travels between bounces is related to thenumber of bounces completed. They shouldrecognize that the data is exponential (that is,there is a common ratio). Students should beencouraged to fit their data to the equationy = ABx where x is the number of bounces and y ishorizontal distance. Ask the students to determinethat A roughly represents the initial horizontaldistance the ball travelled before the first bounce. They could verify this by letting x =0, B0 = 1 and therefore y = A when the number of bounces = 0. Students could alsodrop a basketball or volleyball from a height of 2-3 m and record the successivebounce heights. (A motion detector such as the Calculator Based Ranger [CBR] canbe used.)

C21/F2/F4/C25 Ask students to enter A * B ̂ x into y1 = on the function screen of aTI-83+ graphing calculator. If the initial horizontal distance is 12.5 cm, have studentsset A equal to that value (on the calculator home screen type 12.5 STO A® ® ).Thenhave students try various values for B beginning with B = 1 and press graph tocompare the equation to the data for best fit. Eventually a value for B will result in thebest-fit curve. Have students interpret the B-value and come to realize that it is thesame value as the common ratio.

When students need to find the equation they would be expected to use exponentialregression. Ask students to check their work by finding the exponential equationusing exponential regression.

EXPONENTIAL GROWTH



Exponential Growth

Performance (C21/F2/F4/C25)

1) Any cube larger than 2 2 2× × cmconstructed with cubes will, whendropped in a paint can and removed, havesome of its cubes with three faces painted,some with two faces painted, some withone face painted, and some with no facespainted. For any n n n× × cube (n > 2),complete the table and use it to generatean equation that can be used to determinethe number of faces with:a) no faces painted b) one face paintedc) two faces painted d) three faces painted.

2) Conduct the following experiment to determine how long it takes a cup of boiled waterto lose half its heat.a) boil water and pour into a cupb) insert temperature probec) gather temperature data and create a scatter plotd) fit a curve to the data and find its equation

3) Tower of Hanoi Problem. Use a 1 3× grid and haveeavailable several blocks, each a different size. With oneor more blocks placed in one cell (on top of oneanother in descending order of size—called a tower),the objective is to transfer the tower to another cell inthe minimum number of moves obeying these rules: (1)move only one block at a time to constitute one moveand (2) a block may only be placed on top of a largerblock (or on no block) at any time.a) Complete the table for this game.b) Generate the function that would relate the number of blocks to the minimum

number of moves.

Paper and Pencil (C21/F2/F4/C25)

4) A population growth equation P = 3.8(1.017)t gives an annual percentage populationgrowth of 1.7% for Australia, where P represents population and t represents time inyears.a) Write the annual percentage growth rate of a country whose population is given by P

= 60(1.035)t.b) A country’s population has an annual percentage growth of 6.4%. Its population in

1985 was 53 million. Write an equation to give the population in millions t years from1985.

c) If r is the annual percentage growth, and the initial population is P, write the equationthat gives the population at time t.

5) Kate bought a computer for $3000 to use in a business she is setting up. If it depreciatesat a rate of 30% per year, what will be the depreciated value after one year, two years, ...five years? Find an expression for its value after n years and show it on a graph.Approximately how long does it take for the value of the computer to reduce to half thepresent amount?

dimensions number of faces painted

of the cube 0 1 2 3

1 6 12 83 3 34 4 4

5 5 5

n n n

× ×× ×× ×

× ×M

number minimum

of blocks moves

1 1

2 3

3

4

5


EXPONENTIAL GROWTH




Exponential Growth

A8 demonstrate anunderstanding of theexponential growth ofcompound interest

C26 solve problems thatrequire the application ofcompound interest

B5 demonstrate anunderstanding of and applycompound interest

A8 In today’s financial world, most situations are not calculated using simple interest (I =Prt). Interest is not calculated just once during the life of a loan or investment, but quitefrequently. For example, the mortgage on a house may be compounded semi-annually.Students should understand that if they invest their money in a situation that involvescompound interest, they will be paid interest on their interest. As the money accumulates inthis way, the growth is exponential. Have students explain how they know this.

C26 Students should explore the different ways that compound interest is used in bankingand through investment. This will include some work with annuities (see p. 76).

$100 is invested at 10% per year compounded semi-annually and is cashed after twoyears. Interest will be calculated four times (two times each year). At the end of sixmonths the $100 investment will earn 5% (compounded semi-annually), so that sixmonths later your $105 will earn 5% interest again, growing to $110.25. At the end ofthe fourth interest-earning period (two years) the $100 will have grown to $121.55. Askstudents to create a table and plot the relationship between time periods and totalamount of money. Ask them to explain the patterns they see. What is the common ratio?

B5/A8 Students should learn that the calculation described above can be done moreefficiently using the formula A = P (1 + r) t where t is the number of compounding periods.To help students understand this formula, ask them to find 5% of $100 and then add it to$100. They should get 100 0.05 $5× → , added to ( )$100100 1.05 105× → . Help themunderstand that the (1 + r) is the 105%. (It may help some to show that P (1 + r) is the sameas P + Pr or $100 $100 0.05+ × .) Ask students to explain why A = P (1 + r)t describes anexponential relationship. How is the formula A = P (1 + r)t the same as y = abx? Ask studentsto relate the a and P, and the b and (1 + r).

To help students understand the exponent t, you might have them use the constant featureon their calculators. For example, on the TI-83 have students do the first calculation (firstcompounding period) 100 enter 2nd ans x 1.05 enter , then just press enterfor each subsequent compounding period. Ask students to describe what is happeningand how the number of times they push “enter” relates to the value for t, the number ofcompounding periods.

A8/C26/B5 Students should use the compound interest formula not only to determinehow much an investment will be worth some day, but also how much needs to be invested

now (present value) to provide a certain sum at some future date. ( )11

A

r+

In the study of compound interest students might investigate how long it takes for anamount to double if it is invested at different rates. This should lead to the “rule of 72,”which students can then use to quickly approximate doubling time given a particularsituation. If an amount is invested at 7.2% it will take approximately 10 years to double.Also, if an amount is invested at 10%, it will double in approximately 7.2 years.

x

EXPONENTIAL GROWTH



Exponential Growth

Pencil and Paper (A8)

1) Two situations are given below.

Situation 1: Jeff borrows $1000 from his brother Mike. Mike, to be fair, asksJeff to pay him back the $1000 in one year, plus the interest that accumulates at8% over the year.Situation 2: Sharon borrows $1000 from her banker Bill. Bill tells Sharon thathe can lend her $1000 for a year at 8% per year compounded monthly. Shemust repay the loan in one year.

a) Explain how the debt growth in these situations is different. Explain why.b) Which situation requires more money to satisfy? Explain why.c) Graph these two situations and describe how the graphs differ.

Performance (A8/C26/B5)

3) Islay’s grandmother Sharon gave Islay’s mother $1200 on the day Islay was born.Islay’s mother invested the money at 11.4% per year compounded quarterly.Complete the following steps to see how much money Islay will have on her 18th

birthday.

Performance (A8/C26/B5)

4) Islay decides to cash in the money from the investment (#3) just after her 18th

birthday to use it as a down payment on the purchase of a car. The car she needswill cost $14 595.27 including tax. She borrows the difference at 12.4% per yearcompounded semi-annually over three years. What will be her monthly payments?(Use the Time-Value-Money APPS for this question.)

5) Bobby has just won $50 000 in the lottery. He decides to invest just enough ofthe $50 000 so that in three years he can purchase a $30 000 car. He will investnow at 12.5% per year compounded quarterly. How much of the $50 000 will beleft to buy his math teacher a present? (Ans: $29 262.63)


( )

( )2

a) 0 months = $1200

0.114b) 3 months = $1200 x 1+ = $1200 1.0285 = ____

4

0.114 0.114c) 6 months = $1200 x 1+ 1+ = $1200 1.0 285 = ____

4 4

0.1d) 9 months = $1200 x 1+

( )314 x ________= $1200 1.0285 = ____

4

0.114e) 1 year = $1200 x 1+ x ________ = ____________ = ____

4

0.114f) 2 years = $1200 x 1+ x ________ = ____________ = ____

4

g) 10 years

0.114 = $1200 x 1+ x ________= ____________ = ____

4

0.114h) 18 years = $1200 x 1+ x ________= ____________ = ____

4

EXPONENTIAL GROWTH




Exponential Growth



A1 demonstrate anunderstanding of and applyzero and negative exponents

A2 develop, demonstrate anunderstanding of, and applyproperties of exponents

C11 Now that students have had some opportunity to analyse graphs and use themto explore rates of change, they should take some time to explore other patterns. Inparticular, they should explore the pattern that determines the shape and location ofthe exponential graphs and how that might change as the equation changes. Bylooking at graphs of y = 2x and y = 3x and y = 10x, students should notice that they allpass through the point (0, 1), the “focal” point.

C13/A1/A2 Studying patterns in the graphs and tables will bring up many importantconcepts dealing with number sense. For example,

i) In the paragraph above students should be able to generalize that any base tothe exponent 0 will result in 1.

ii) In creating the table for y = 2x, when x = –1, –2 , y will result in fractions

1 1,

2 4, ... Instead, students may use their calculators and get y-values of 0.5,

0.25 ...

Upon further investigation, they should notice that 1 12

2− = , and 2 1

24

− = , and

1 13

3− = , and

2 13

9− = . Pronouncing decimals as fractions will help students make the

connections. For example, when students see 0.5, they should say this as “five-tenths,” and 0.25 is pronounced “twenty-five hundredths.” All of this will be visuallyreinforced as they find these corresponding values on the graphs. Students should beable to describe these equalities in words, and they should be encouraged to view

numbers in their various representations 1 1

10 0.110

− ⇒ ⇒ . This is a logical

connection with the approach taken in earlier grades but needs to be reinforced andextended. Mental math activities should reinforce understanding.

Students should look closely at the generalized equation y = a x and explore whathappens to its graph:

a) wheni) a = 1 iv) a < 0ii) 0 < a < 1 v) a > 1iii) a = 0

b) Students should graph y = 2 x and trace for y-values when

i) x = 0 iv)1

2x =

ii) x = -1 v)13

x =

iii) x = 1

(A table might be helpful)

EXPONENTIAL GROWTH



Exponential Growth

Mental math (C11/C13/A1)

1) Match each graph in the left column to a given description in the right column, ifpossible.

Graph Stimulus

a) i) On the first day I received one cent, the second day two cents, the third day four cents, the fifth day eight cents, ... [Ans: (c) ]

ii) y = bx, where b is greater than 0 and less than 1

[Ans: (a) ]

b) iii) y = 2–x [Ans: (a) ]

iv) I poured myself some coffee that was too hot to drink, so I let it cool. I fell asleep. [Ans: (b) ]

v) 32

x

y = [Ans: (c) ]

vi) [Ans: (a) is best match ]

c)

d)

Pencil and Paper (C13/A1/A2)

2) Explain how you would know that if 1 12

2− = , then

12 33 2

− =

.

3) Use the graph y = 4x to explain why 40 = 1.4) Evaluate:

i)24

5

−

ii) 30 + 2 –2

x –3 0 5 6

y 30 10 5 4.99


EXPONENTIAL GROWTH




Exponential Growth

A2 develop, demonstrate anunderstanding of, and applyproperties of exponents

A1 demonstrate anunderstanding of and applyzero and negative exponents

C5 determine and describepatterns and use them to solveproblems

A2/A1/C5 Students may already have realized from their work with compoundinterest and money growth that, in the equation A = 750 (1.05)n, whenn = 0, (1.05)0 = 1, and since they have not yet invested their money (n = 0), they stillhave the $750 they started with.

A1 Zero and negative exponents can be explored using patterns when exponentialfunctions are studied. For example, complete the table below for y = 10x bygeneralizing the pattern observed.

104 = 10000 10-1

103 = 1000 10-2

102 = 100 10-3

101 = 10 rule? 10-4

100 =

As students explore negative exponents, they should realize that a number with anegative exponent can always be written with a positive exponent, but one form is not“better” than another, just different. Traditional teaching of exponents has left theimpression that a base raised to a negative exponent should be changed to a baseraised to a positive. In many contexts (e.g., scientific notation) the negative exponentis preferred.

C5 Students have learned in previous courses that scientific notation can be used torecord numbers using proper significant digits. For example, to express the number2900 (two significant digits) with three significant digits, use this form: 32.90 10× .An effort should be made to remind students of the patterns for scientific notation.Discussion: “If you want to express 385 as a number in scientific notation, then you

could write 385 as 385

100100

× , which simplifies to 23.85 100, or 3.85 10× × .”

EXPONENTIAL GROWTH



Exponential Growth

Pencil and Paper (A2/A1)

1) Use the graph of y = 3x to explain why:

a)2 1

39

− = b) 30 = 1 c) 0.53 1.732B

2) Rewrite in another form:a) 2–3 with a positive exponentb) 3120

c) 1 as a power of 10.

d)1

8 with a base 2

e) 3 250 000 in scientific notation

3) Explain how 625 can be expressed using base 5.

Mental math (A2/A1)

4) Expressa) 32 with a base 2 d) 2–3 with a positive exponentb) 27 with a base 3 e) 47 with a negative exponent

c) 16 as a power of 2. f)

512

− as a whole number

5) Solvea) 4x = 16 b) 3x = 9 c) 2(5x) = 50 d) 3–3 = x

Performance (A2/A1/C5)

6) Greg loses points on a test for saying that 81

2

−

equals –28 . Tell Greg what error

he made, and how to avoid it in the future.

7) Divide 14

7

6.42 102.91 10−

××

using estimation. Explain your solution.


EXPONENTIAL GROWTH




Exponential Growth



B6 determine the amountand present value of annuities

C26/B5 Students should solve a variety of problems using various techniques that involveexponential equations. Several of these contexts have been referred to in earlier elaborationsuch as compound interest, the allowance problem, and population growth.

Compound interest should be used by students to determine:

€ the amount of an investment over a period of time with different compounding periods.For example, calculate the amount of money Amanda will have at age 18 if $5000 isinvested at 10% per year, compounded monthly when she is one year old.

( )( )12 170.1$5000 1

12A

= +

€ the present value of a loan payment. For example, Toby decides to invest enough moneynow so that in 3 years, he can help his daughter buy a car. He wants to give her $3000.How much must he invest now at 12.5% compounded semi-annually?3000 = P(1 + .0625)2(3)

6

3000

1.0625P =

B6 An annuity is a sequence of payments made at equal time intervals. The amount of theannuity is the sum of the sequence of payments, including all the interest earned. Examplesof annuities include mortgage payments, pension cheques, and payments made to repayloans. Annuities can be illustrated visually with a time diagram. Consider Wibur’s situation:

Wilbur wishes to save money for a stereo system. He plans to set aside $50 per month,beginning at the end of January, and to invest his money in a savings program thatpays at 12% per year compounded monthly. Wilbur’s last payment will be made atthe end of December.

Note that Wilbur’s first payment is at the end of the first payment period. Wilbur’sJanuary payment will earn interest at 12% per year, compounded monthly, for 11months. Using the compound interest formula

( ) ( )11 11

1

50 1 0.01 or 50 1.01

nti

A Pn

= +

= +

Similarly his second payment will be 50(1.01)10, and these payments for asequence of values all the way to the December payment of 50(1.01)0, or just$50.Students can be given the formulas for interest, annuities, etc. when in testingsituations in this course.

... continued

EXPONENTIAL GROWTH



Exponential Growth

Performance (C26/B5)

1) Dorothy buys a $15 000 car. She is able to pay $3000 as a down payment. Sheborrows the rest of the money from a bank at 10.25% per year, compounded semi-annually over five years. If she makes payments at the end of each month, what isthe total amount she will pay for the car? [Ans: $18 311.24 ]

2) Ronnie would like to have $8500 available when he graduates from high school tohelp expand his business. How much should he invest now at 12% per year,compounded monthly, in order to have the $8500 two years from now?

3) Mr. and Mrs. Maze wish to give their newborn son a cheque for $20 000 on his21st birthday. How much money must they invest at his birth if the money willearn 4.8% per year compounded semi-annually.a) Read Arthur’s solution and attempt to find any errors.b) Explain to Arthur what he did wrong and how to fix it.

Arthur’s solution begins ... ( )21

20000 1 .048

2A

+=

Performance (C26/B5/B6)

4) The following time diagram represents the amount of an annuity with a term offive years:a) What is the payment being made?b) What is the annual interest rate?

c) What is the compounding period?

5) Cindy plans to save for her new baby that she plans to have in three years. Shedecides to invest $1000 twice a year at 9.8% per year, compounded semi-annually.Draw a time diagram to represent her investment.

6) A deck can be built for $2 300. How much should you start investing every monthif the interest rate is 8% per year, compounded monthly, and you hope to haveenough cash in 18 months?


EXPONENTIAL GROWTH




Exponential Growth



B6 determine the amountand present value of annuities

B6 The same pattern will continue for all Wilbur’s payments. Wilbur’s 11th or second-last payment will be made at the end of November. This $50 payment will earninterest for only one month, so it will amount to 50(1.01)1. His final payment willearn no interest, since he plans to take all the savings and buy the stereo.

Each of his payments earns interest except the last, and the accumulated sum,including interest, becomes the total amount he receives. To find the total amount,students should add the 12 payments

50(1.01)0 + 50(1.01)1 + 50(1.01)2 + ... + 50(1.01)12 .

B6/C26/B5 Students should be encouraged to use the Finance feature on the TI-83calculator to find the sum, in addition to adding the 12 payments as in the exampleabove. Using the Finance feature, students will select 1:TVM Solver ... and entervalues in appropriate places. N is the total number of payments (12 in the aboveexample). I% asks for the interest rate as a percentage. PV is the present value (zero inthis case since Wilbur begins with no money). PMT is the payment amount ($50 inthis case). FV is the future value, which is the sum of all the payments andaccumulated interest (the sum of the 12 payments in the example above). Since this isthe amount students need to find, they will leave the value as zero and come back tothis in a moment. P/Y is the number of payment periods (in this case 12), and C/Y isthe number of compounding periods (in this case 12). The payments are at the end ofthe month. Now return the cursor to the FV, and press “Solve” (2nd Enter) tocalculate the sum, or future value.

A variation on the annuity problem would happen when a future value is given andthe present value is asked for. For example:

Lucy and Pierre are saving money to celebrate their parents’ 50th anniversary inthree years. They would like to have $5000 for the occasion. What would theyhave to invest now at 10% per year, compounded monthly, to reach theirobjective?5000 = P(1 + .083)36

P = 5000/(1.083)36

EXPONENTIAL GROWTH



Exponential Growth

Performance (B6/C26/B5)

7) The following time diagram represents the amount of an annuity with a term offive years:

a) How does the diagram help you to recognize that this represents an annuitysituation?

b) What is the periodic payment?c) What is the annual interest rate?d) How many payments are made?

8) Betty Lou begins to work part-time when she turns 16 and plans to begin thecourse at age 19. She is looking to her future costs of education. She wants toknow how much money she will need to set aside at the beginning of each monthat 10% per year, compounded monthly, in order to have $10,000 for first year ofstudies in an information technology course. [Ans: $237.36 ]

9) Complete the time diagram.The interest is 7% per year, compounded annually. Payments of $1500 are madeannually for 10 years.

10) Cindi plans to save for community college in three years. She decides to invest $1000 twice a year at 15% per year, compounded semi-annually. Draw a time diagram to represent her investment.

11) Joanne and Sheila set up an annuity so that in 12 years they will have enough money to purchase a new tractor for their horse farm. They deposit $4750 at the end of every six months in an account that earns 11% per year, compounded semi-annually. Set up a time diagram to illustrate their situation.

12) The Allens are interested in buying the house listed in the following ad:Saint John, 2 storey, 3 BRs, centre hall plan, immaculate decor, hardwood floors,ground-level family room, and new kitchen. Huge mature treed lot. Asking$264 800.

They have a down payment of $150 000 and would take a mortgage with theirbank at 12.5% per year, compounded monthly, and amortized over 25 years.a) What is the value of the mortgage?b) What is the monthly interest rate?c) How many mortgage payments will they have to make?d) What are their monthly payments?


EXPONENTIAL GROWTH




Exponential Growth

Unit 4

Circle Geometry

(30 Hours)

CIRCLE GEOMETRY




Circle Geometry

E1 perform geometricconstructions and analyse theproperties of the resultingfigures

E5 apply inductive reasoningto make conjectures ingeometric situations

E10 present informaldeductive arguments

E6 explore, make conjecturesabout, and apply centres ofcircles

E7 explore, make conjecturesabout, and apply chordproperties in circles

E8 explore, make conjecturesabout, and apply anglerelationships in circles

E1 Geometry is a rich field of mathematical study. The world around us is inherentlygeometric, and humankind’s creations most often reflect geometric principles. Theconcrete and visual nature of geometry resonates with certain learning styles, andgeometry’s pervasiveness in our environment facilitates connecting the study ofgeometry to meaningful situations. This is as true for circle geometry, the focus of thisunit, as for geometry in general. Whether determining the correct location for handleson a bucket, finding the centre of a circle in an irrigation project, or determining thelength of a tangent to the earth from an orbiting satellite, properties of circles (andlines, line segments, and/or angles associated with them) come into play.

E5/E10 Geometric figures such as segments, lines, angles, polygons, circles, andplanes are each sets of points that are subsets of the universal set called space. Insynthetic (Euclidean) geometry, these geometric figures can be drawn anywhere on aplane in space; in analytical (coordinate) geometry, a reference system is added, andimportant points on the figures are assigned coordinates. Using transformations,these figures, with or without coordinates, can be moved in space by followingspecific rules. In all perspectives students seek to discover patterns among figures orwithin a fixed figure.

Students need many opportunities to explore geometric situations, look for commonelements (or patterns) in them, and make appropriate conjectures. They also need toreach an understanding that, while this inductive process of observing multiple casesand conjecturing seems to imply the truth of a relationship, deductive reasoning isrequired to establish the truth of any conjecture in general. As part of this process,students should also realize that measurements with tools i) are not accurate and ii)deal only with specific cases and are, therefore, not adequate as proofs.

Students should be exposed to informal proof, with the understanding that a logicalargument can take many different forms. This unit provides the opportunity forstudents to present informal deductive arguments.

E6/E7/E8/E5/E10 In particular, contexts will be explored, and theorems conjectured,proven, and applied, with respect to chord properties in circles, inscribed and centralangle relationships, and centres of circles. The treatment of these circle topics is notintended to be exhaustive, but is determined to a significant extent by the contextsexamined.

CIRCLE GEOMETRY



Circle Geometry

Activity (E1/E6/E5)

1) a) Begin with a circle of any size (given to the student).i) Fold the circle in half make a crease.ii) Open up the circle and fold in half differently.iii) Open up the circle and investigate the intersection point.iv) Compare results with classmates.v) Make a conjecture; test the conjecture.

b) Begin with a circle of any size (given to the student).i) Make a fold anywhere on the circle—make a crease.ii) Repeat the first step with a second fold and crease.iii) Mark the first fold AB at its end point, the second CD.iv) Fold A onto B make a crease.v) Open the circle, fold C onto D make a crease.vi) Investigate the intersection of the last two creases.vii) Compare results with classmates.viii) Make a conjecture; test the conjecture.

Journal (E10/E8)

2) Students have been asked to make the logical argument that an angle subtended ina semicircle is 90°. Mary Beth wants to use the relationship between inscribed andcentral angles. Explain how she would use this relationship in her arguments.

Journal (E10/E7)

3) In the diagram the segments AB and CD are chords of thecircle with centre O. Describe how you would convince yourpartner that AB > CD without measuring the segments ABand CD.


Curriculum and EvaluationStandards for SchoolMathematics (NCTM, 1989)

Geometry from MultiplePerspectives. Addenda Series,Grades 9-12, (NCTM, 1991)

Computer Software

1) Richard Parris Wingeom(free from http://mathexeter.edu/rparris)

2) Geometers Sketchpad

3) APPs for the IT-83 Plus -Cabri Jr (free from http://education.ti.com)

CIRCLE GEOMETRY




Circle Geometry



E1/E5 If students begin with a large isosceles triangle, theyshould explore

a) how to construct the circumcircleb) how to construct the incircle

These require students to investigate the intersection points ofperpendicular bisectors and angle bisectors. They should explorethe construction of incircles and circumcircles for scalene and obtuse triangles.

After their exploring, students might conjecture

€ The three perpendicular bisectors of the sides of any triangle intersect at one point,and this point is the centre of the circumscribed circle.

€ The three angle bisectors of any triangle intersect at one point and this point is thecentre of the inscribed circle.

Geometry software affords students the opportunity to investigate geometricproperties inductively. A more universally available and equally interesting opportunityexists with Patty Paper. (Patty Papers are the small squares of paper that are used bysupermarkets to seperate hamburger patties) With a minimum of instruction and veryfew definitions and assumptions, students can perform the six basic Euclideanconstructions, go on to investigate properties of points of intersection of medians,altitudes, angle bisectors, and perpendicular bisectors within triangles and among theextensions, and discover centres of inscribed and circumscribed circles. This emphasison inductive reasoning is very important as it keeps the geometry alive and forcesstudents to be clear and accurate in their use of appropriate language and symbolismin geometry.

CIRCLE GEOMETRY



Circle Geometry

Performance (E1/E5)

1) Construct by paper folding the three perpendicular bisectors in each of thefollowing triangles. What conjecture might you make about the three perpendicularbisectors of a triangle? [The point of intersection is the centre of the circumcircle.]

2) Construct by paper folding the three angle bisectors in each of the abovetriangles. What conjecture might you make about the three angle bisectors of atriangle? [The point of intersection is the centre of the incircle.]

3) Given the ABCV with AB = AC. Construct the perpendicular bisector of BC and

the angle bisector of BAC∠ . Do the same for two otherdifferent isosceles triangles. What conjecture can you make.Explain how you might test your conjecture. Why do youbelieve your conjecture is correct?n [The bisector of the vertexangle is the altitude to the base or is a bisector of the base.]

4) Using the same triangle ABCV as above, fold the triangle on the perpendicularbisector that you constructed. Where is the image of C? Do you think this will betrue for all isosceles triangles? Explain.

5) Given ABCV . Bisect ABC∠ . Pick any point on the bisector and drawperpendiculars to the two sides of the angle ABC∠ . Pickanother point and do the same. Make a conjecture aboutthese two perpendiculars. Explain how to test yourconjecture.[Any point on the angle bisector is equidistantfrom the sides/arms of the angle.]




Computer Software




⊥

CIRCLE GEOMETRY




Circle Geometry

D2 determine midpoints andthe lengths of line segmentsusing coordinate geometry


D2 Students should understand that the determination of midpoint and length of anoblique line segment cannot be read from a graph or picture as readily as they can forhorizontal and vertical lines. Students can generalize from activities such as on p. 85“Standards—A Core Curriculum,” Addenda Series, (NCTM) to a formula formidpoint and distance (see next page).

In the activity on the next page, teachers should ask students to respond to the twoquestions above the graph in the first paragraph. The teacher might provide diagramsfor each situation and have students orally discuss how they might find the middle of

the goal line ( )AB or how to determine how far the ball is from the middle of the

goal line. Parts (a), (b), (c), and (d) in the third column are designed to help lead thestudents to be able to better answer the questions being discussed.

For example, finding the lengths AC and BC on the graph in column three are simplefor students, since they just have to count from A to C to get the length AC and fromC to B to get the length CB. Part (b) assures that students note the right angle andthus in (c) can use the Pythagorean relationship to find length AB. In (d) studentsshould find midpoints of AC and BC by ‘going halfway’. Teachers should help

students connect the “halfway” coordinates with 1 21 2 ,

2 2

y yx x ++ = (mean of x-

coordinates, means of y- coordinates), so that students can use this midpoint formulato find the midpoint of AB.

When students attempt (e) remind them how they found the length of AB using thePythagorean Theorem. The teacher then needs to lead them to see that thecoordinates can be used to get the same result. By subtracting the x values for A and Bthey get the length AC, and by subtracting the y values they get length BC; thus

developing the distance formula: ( ) ( )222 1 2 1d x x y y= − + − . Part (e) provides

another opportunity to make that connection.

D2/E10 Students could practise the use of the distance formula and midpoint formulain the context of the questions and activities from previous pages. For example, from1) on page 83 have students use the distance formula to check their conjecture thatthe point of intersection from the three perpendicular bisectors of the three sides ofany triangle will be equidistant to the three vertices of the triangle. This point is thecircumcentre of the triangle. Give them the coordinates for the three vertices and thecentre of the circle and have them check the length of the three radii.

CIRCLE GEOMETRY



Circle Geometry

Activity (D2)

1) You are a computer game designer. In the game you are designing, a player needs totoss a ball to land as close as possible to the centre of a goal line. Sometimes thegoal line is horizontal, but not always. You know that the computer screen is madeup of pixels that can be assigned coordinates. How can you find the midpoint ofany goal line, and how can you determine thedistance of the player’s ball from the centre of thegoal line?a) How long is AC? BC?b) What kind of angle is <ACB?c) Find the distance AB.

d) Find the midpoints of AB, BC, AC.

e) Create a right triangle to help you find the length of RS in each diagram.

(i) (ii) (iii)

2) Ask students to find the midpoints of AB and AC , thenask them to find the length of the segment joining thesepoints. Ask students to compare its length to the lengthBC and to describe what property they have examined. Askstudents to discuss whether they have proved this or not.[Length of Segment = (BC) ]




Computer Software




1

2

S

R

(6,4)

(2,1)

R

S (6,2)

(1,7)

CIRCLE GEOMETRY




Circle Geometry


E6 explore, make conjecturesabout, and apply centres ofcircles


E2 describe and applysymmetry


E1/E6/E5 Students need to begin their study of circles by exploring patterns andmaking and verifying conjectures. They might begin their exploration with an activitylike the following:

Activity

€ On a blank sheet of paper (or using technology) place any two points P and Q.Construct a circle that passes through P and Q such that PQ is not the diameterand explain how you located the centre (C). What kind of a triangle must PQCbe? Explain.

€ Construct three other circles that pass through P and Q. Name their centres D, E,and F.

€ Fold P onto Q, making a crease to indicate the fold line.€ What do you notice about the points C, D, E, and F?€ Name the point where the crease intersects P, Q, as M. Is M the midpoint of PQ?

Justify your answer.€ Is PQ ⊥ to the fold line? How do you know?

€ Make a conjecture. Test your conjecture.€ Take any point A on the fold line, join it to P and Q. Make a conjecture. Test your

conjecture.

E1/E6/E5/E2 While exploring patterns (as in the previous activity), students mightuse paper-folding techniques and/or measurement tools like rulers, dividers,compasses, and protractors. In so doing they will be using both transformational andEuclidean techniques. They will also be observing lines and points of symmetry, andthey should be expected to describe the symmetry and how it defines certain figureslike isosceles triangles and rhombi. They will also be using inductive reasoning tomake conjectures such as

€ Any point that is equidistant from two points on a circle must be on theperpendicular bisector of the chord joining those two points, or its converse

€ Any point that is on the perpendicular bisector of a chord of a circle must beequidistant from the end points of that chord.

E1/E10 Teachers will need to model the thinking processes necessary to generateinformal proofs. As well, it may well be necessary to spend time reaquainting studentswith the geometric properties with which they are already familiar (e.g., congruenttriangles, angle sum of a triangle, vertically opposite angles, parallel line theorems).

Consider passing out transparencies to students copied from ablackline master such as the diagram shown. Have each studentor group locate at least one point that is equidistant from theendpoints of the chord. Overlay the transparencies on theoverhead to see that the points form the perpendicular bisectorof the chord.

CIRCLE GEOMETRY



Circle Geometry

Activity (E1/E6/E5/E2/E10)

1) a) Students begin with an isosceles triangle. Ask them to construct twoperpendicular bisectors, and label the intersection point P. Have them measurethe distance from P to the three vertices. Ask them to draw a conclusion basedon their measures. [Answer: P is equidistant to the vertices.]

b) Using a compass have students construct a circle with radius “the distance fromP to one vertex”. This circle is called the circumscribed circle or circumcircle ofthe triangle. Have students explain what they think circumscribed means.

c) Students start again with another isosceles triangle. Have them bisect the twobase angles of the isosceles triangle by paper-folding methods and name theintersection point Q. Through Q have them fold the paper so that one base-angle vertex maps onto the other. Ask them to describe what must be trueabout the angle formed by the paper fold and the base of the triangle. Call thispoint M [a 90-degree angle].

d) Again, using paper folding, have students bisect the third angle and describewhat points the crease in the paper includes [includes Q, and P, and M]. Askstudents if they think this will be true for all types of triangles. Have them testwhat they think on different triangles [on all triangles it will include Q].

e) Folding through Q, have students fold the third vertex onto each of the basevertices, making a crease each time. Have them name the intersection points(the crease with each side of the triangle) N and S. Ask them what they thinkwill be true about QM, QN, and QS [same length]. Have them check theirpredictions.

f) Using QM as radius, have students construct a circle, centre Q. This circle iscalled the incircle of the triangle. Have students explain what an incircle is.

Activity (E1/E6/E5)

g) Have students explore the two centre points (the circumcentre and the incentre)and describe what they notice about them. Have them include discussion aboutwhy this might be true on this triangle.

h) Have students describe the symmetry in the triangle, and include in theirdiscussion the terms bisector of the vertex angle, the perpendicular bisector ofthe base, median, and altitude.

i) Have students examine the intersection points that occur when they draw thethree medians and the three altitudes of a triangle. Have them compare thesepoints of intersection with points P and Q, using isosceles triangles as well asother triangles. Students should discuss how symmetry plays a role in all of theabove conjectures, and describe the special characteristics of symmetry inisosceles and equilateral triangles.

j) Have students solve this problem: “Mary and Bill both want their inscribedcircle to pass through the centre of their circumscribed circle. How can theyachieve this?” [Ansl: Construct an isosceles right triangle]

Note: This is a good problem to explore using geometry software.




Computer Software




CIRCLE GEOMETRY




Circle Geometry


E7 explore, make conjecturesabout, and apply chordproperties in circles



D2 determine midpoints andthe length of line segmentsusing coordinate geometry

E1/E7 While students are considering the shape of their isosceles triangle, theopportunity may arise for them to explore the relationships between chord length andnearness to the centre of the circle.

Activity:

a) Begin with a circle (make your own) and mark the centre point.€ By folding, create five chords (creases) of different lengths.€ Fold one end of each chord onto itself—make a crease.€ Investigate the lengths of these creases from the centre of the circle to the

chord—make a conjecture.

b) Begin with any circle (make your own) and mark the centre point.

€ Make five folds creating 5 chords all of equal length (fold into the centre).€ Investigate the distance that each is from the centre—make a conjecture.

E1/E7/E5 From (a), students might conjecture that the longer the chord is, the closerit is to the centre; from (b), chords of equal length are the same distance to the centre.

E10/D2 Students might use coordinates to find lengths andmidpoints to help reach these conjectures. For example, bysuperimposing a coordinate system, students can calculate themidpoints of AC and BC and the distances for themidpoints to the centre.

A B

C

CIRCLE GEOMETRY



Circle Geometry

Performance (E1/E7/E5/E10)

1) Construct two circles with the same radius r so that each circle passes through thecentre of the other circle. Label the centres P and Q, and construct the segmentPQ. The two circles intersect at A and B.a) What is the relationship between the segments AB and PQ? Explain your

thinking. [AB PQ ]b) Explain how you might prove your conjecture in (a).

2) Construct a large circle and two non-parallel congruent chordsthat are not diameters.a) Compare their distances to the centre of the circle. [They

are the same.]b) Write your findings in (a) as a conjecture.c) Test your conjecture on other circles.d) Explain how you might prove your conjecture.

Performance (D2)

3) Do question 2 above one more time, this time on a piece of graph paper. Usecoordinates to find distances. Start with the centre at (0, 0).

4) Given that A (7, 9) and B (–3, –5) are the endpoints of the diameter of a circle,show that C (2, 2) is the centre. Determine if thepoint D (9, 7) is on the circle. Explain.

5) Given the diagram, and the coordinates for A, B,C, and D, ask students to find the midpoints of

AB and CD . Have them explain why this point

must be the centre of the circle.

Performance (E1)

6) Use a circular object to trace a circle onto yourpaper. Without using a compass, locate the centre of thecircle. (Use a compass to check your work.)

Performance (E1/E7)

7) A piece of circular plate was recently dug up on an islandin the Mediterranean Ocean. The discoverer of the platewishes to calculate the diameter of the original plate.Describe how he could do this.




Computer Software




⊥

CIRCLE GEOMETRY




Circle Geometry


E8 explore, make conjecturesabout, and apply anglerelationships in circles


E1/E8/E5 While exploring ideas and relationships within circles, some students madethe following conjectures:

a) The central angle marked 1 is twice the measurement ofthe inscribed angle marked 2, and they both are connectedto end points of the same arc, marked 3.

b) Inscribed angles (1, 2, 3) that share the same arc, marked4, all have the same measure.

c) Angles inscribed in semicircles are right angles.

d) When the centre of the circle is joined to a point where asegment touches the circle, a right angle is formed.

CIRCLE GEOMETRY



Circle Geometry

Activity (E1/E8/E5)

1) a) Draw any circle. Select any minor arc »AB. Join the endpoints of the arc to the

centre of the circle. Pick any point P on the major arc »AB. Join the endpoints

of the arc »AB to the point P. What conjecture can you make with respect to

the relationship between the measurements of the two angles? [The centralangle = 2 (inscribed angle) when subtended by the same arc.]

b) Pick two other points R and S in the major arc »AB. What new conjecture can

you make with respect to the angle measures. [The inscribed angles subtendedby the same arc are equal. In this case, ]

c) Construct another circle with diameter PQ . Select any position R on the

circumference. Measure the angle PRQÐ [ = 900]. Do you think you

will get a different answer depending on where you select the point R? Explain. d) Begin with a circle on a rectangular sheet of paper.

i) Make a fold at three different points on the circumference to produce creases that touch the circle at only those three points. [These creases will be tangents to the circle.]ii) Join the points to the centre and investigate the angles formed between the radii and the tangents—make a conjecture. [A radius is perpendicular to a tangent at the point of tangency.]

e) Answer the following questions based on yourconjectures. O is the centre of the circle.

i) if ARM 32Ð = ° , then AOM Ð = ° [640]

ii) if SA is tangent at A, then SAO Ð = ° [900]

iii) if ARM 44Ð = ° then APM Ð = ° [440]

iv) Draw the angle AMDÐ , what is its measure?Explain. [900. It is inscribed in a semicircle.]

Performance (E1/E8/E5)

2) Using geometry software,a) Construct a circle, with any two chords that do not intersect

within the circle.b) Extend the chords to intersect outside the circle.c) Make a conjecture relating the measure of the angle formed

outside the circle by the intersecting chords and the measures of the twointercepted arcs. [The angle is equal to half the difference of the measures of thetwo intercepted arcs.]

Performance (E10)

3) O is the centre of the circle shown at the right above. The two central angles arecongruent. Write a argument that would convince your friend that AB = MN.




Computer Software




PRQÐ

APB = ARB = ASB.Ð Ð Ð

CIRCLE GEOMETRY




Circle Geometry


E10 As students make conjectures and test their validity, they should be encouragedto go beyond testing the conjectures by explaining or trying to verify them withlogical, deductive reasoning.

Students should understand that logical arguments need to be presented in anorganized way to prove that their conjectures are true. However, at this level, thedeductive arguments can be presented informally. For example, the following scenariomight take place:

In Mr. Doodle’s math class, a group of students were trying to prove the conjecturethat the perpendicular bisector of a chord passes through the centre of a circle.a) James said, “Oh, that’s easy! Just fold one end of the chord onto the other,

make a crease, and see if it goes through the centre.” Michelle said, “Yeah,James, that would work, but it doesn’t prove it.” Kenny jumped in, saying “Ifthe crease is the perpendicular bisector of the chord, doesn’t that mean that anypoint on it is equidistant to the end points of the chord?”

b) Jennifer’s proof begins: “The chord lie on the x-axis with end points A (–5, 0)and B (5, 0). Then the y-axis is the perpendicular bisector at (0, 0), themidpoint of AB.” Then, she asked the class, “If the radius of the circle is 10,find the centre. If the radius is 7, find the centre. What conclusion can youmake about the centre and the perpendicular bisector?”

c) Malcolm asks, “What if we’re given coordinates for the endpoints of the chord

AB , and for the centre, O, of the circle—could we prove it then? Couldn’t wee

just say that if the distances OA and OB are equal, then O is on theperpendicular bisector of AB since the triangle OAB is isosceles, and anisosceles triangle has one line of symmetry, which is a perpendicular bisector.

Angle measurement can be used to deduce properties andrelationships. For example, students could be asked to find arelationship between the angle COA and the angle CBA in thefollowing diagram. Students would say that m B m A 50Ð + Ð = °because of the exterior angle relationship.

They then would deduce m B 25Ð = ° and is half of the measure

of angle COA because the OBAD is isosceles (equal radii).

CIRCLE GEOMETRY



Circle Geometry

Performance (E10)

1) The d’Entremonts’ backyard is the shape of a quadrilateral marked by corners thatcould be located on a Cartesian Plane at A (0, 5), B (0, 0), C (12, 0), and D (9,8). Jean thought if she tied string to the midpoints of each side he would have arectangle. Marie thought it would be a square. Were either of them correct?Explain.

2) Sarah draws a diagram like this and says that she conjectures that some triangles arecongruent. She writes the following statements to present her argument:

If I fold on PM , O will fold onto itself and A has to landon B. So, the AOMV fits perfectly on BOMV . They’rerecongruent.

Do you agree that this proves that the triangles arecongruent? Explain.

3) To prove that the inscribed angle is half the central angle,Andrew gave the following argument:

Since AO, BO, and CO are equal radii of a circle, thenABOV and BCOV are isosceles so that the base angles

have equal measures.a = 20° + 20° = 40°b = 30° + 30° = 60°So a + b = 100°but m ABC 20 30 50Ð = + =So the central angle is half the inscribed angle.

Celia said that this does not prove for all angle measures,only for those given in the diagram (1). Use diagram (2) toprove for all angle measures.

4) Given any isosceles triangle, explain why you think that the intersection of thebisectors of the base angles would be the centre for the inscribed circle.

Performance (E10)

5) Caleb conjectured that every triangle has an inscribed circle. Henry does not thinkhe is correct. How could you convince Henry that Caleb is correct?

6) a) Show by construction that the three medians of a triangle intersect at onepoint.

b) Do you think this will always be true? Test yourconjecture on this triangle.

c) Do you think that the intersection of the medians ison the same line as the centres of the inscribed andcircumscribed circles of any triangle? Explain.




Computer Software




CIRCLE GEOMETRY




Circle Geometry

Unit 5

Probability

(15-20 Hours)

PROBABILITY




Probability

G2 develop an understandingthat determining probabilityrequires the quantifying ofoutcomes

G3 demonstrate anunderstanding of thefundamental countingprinciple and apply it tocalculate probabilities

G2 Every day, students experience a variety of situations. Some involve makingdecisions based on their previous knowledge of similar situations.

€ Should they do their math homework tonight or during their spare period beforemath class tomorrow?

€ Should they challenge a friend to a game of racquetball or checkers?€ Should they buy a ticket on a car raffle?€ Should they take their umbrella today?

Before making the decision, what they must ask themselves is “What is the chance ofthis decision working out in my favour?”

In probability, events are given numbers ranging from 0 to 1, where 0 refers to thingsthat never happen and 1 refers to things that always happen.

In their previous studies (grades 7–9) students have created and solved problemsusing probabilities, including the use of tree and area diagrams, and simulations.They have compared theoretical and experimental probabilities of both single andcomplementary events, and dependent and independent events. Theoreticalprobabilities are those that result from theory (what should happen mathematically),while experimental probabilities are those that result from experiments or repeatedtrials of performing the event. Students also have examined how to calculate theprobability of complementary events. The probability of an event happening and itscomplement add to make 1. They also study two independent events, A and B, wherethe probability of A and B is equal to ( ) ( )P A P B× .

G2/G3 Sometimes the task of listing and counting all the outcomes in a givensituation is unrealistic because the sample space may contain hundreds or thousandsof outcomes.

The fundamental counting principle enables students to find the number ofoutcomes without listing and counting each one. If the number of ways of choosingevent A is n(A) and the number of ways of choosing an independent event B is n(B),

then ( ) ( ) ( ) and then n A B n A n B= × , and n(A or B) = n(A) + n(B). The first is the

multiplication principle, the second, the addition principle.

Sometimes events are not independent. For example, suppose a box contains threered marbles and two blue marbles, all the same size. A marble is drawn at random.

The probability that it is red is 35 . If the marble is then replaced, the probability of

picking a red marble again is 35 . However, if it is not replaced, then when another

marble is picked the probability of it being red is now 25 . The probability for the

second selection of a marble is dependent on the first selection not being returned tothe box.

... continued

PROBABILITY



Probability

Activity (G2/G3)

1) Two students are playing “grab” with a deck of special “grab” cards. One studenthas a triangular shaped deck with 16 ones, 12 twos, 8 threes, and 4 fours. Theother has a rectangular-shaped deck with 10 each of ones, twos, threes, and fours.The decks are well shuffled. One person turns over the top card of the triangulardeck while the second person turns over the top card of the rectangular deck. A“grab” is made when two cards match (a double).a) There are 40 cards in each deck. What is the total number of pairs of cards

which could be played? [Ans: 32, since there are at most 10 matches for ones,10 matches for twos, 8 matches for threes and 4 for fours.]

b) How many of these pairs are “double ones”; that is, a one from the triangulardeck and a one from the rectangular deck? [Ans: 10]

c) How many are double twos? double threes? double fours?d) For equally likely outcomes, the probability of an event is “the number of

outcomes that correspond to the event” divided by what?e) So, the probability of a double one is “what” divided by “the total number of

pairs”?f) Use the multiplication principle and your answers to (c) to find the probability

of i) a double one, ii) a double two, iii) a doubleg) A circular deck has 10 ones, 20 twos, 10 threes, and no fours. Calculate the

probability of a “grab” if a triangular deck is played against a circular deck.

Performance (G2/B3)

2) Telephone numbers are often used as random number generators. Assume that acomputer randomly generates the last digit of a telephone number. What is theprobability that the number is:a) an 8 or 9 ? [Ans: ]

b) odd or under 4 ? [Ans: ]

c) odd or greater than 2 ? [Ans: ]

3) A airplane holds 176 passengers, 35 seats are reserved for business travellers,including 15 aisle seats, 40 of the remaining seats are aisle seats. If a late passengeris randomly assigned a seat, find the probability of getting an aisle seat or one inthe business travellers’ section.

4) Use the given table, which represents the number of peoplewho died from accidents and their respective ages, and ineach case assuming that one person is selected at randomfrom this groupa) Find the probability of selecting someone under 5 or

over 74.b) Find the probability of selecting someone from 15 to

64.c) Find the probability of selecting someone under 45 or

from 25 to 74.


Computer Software:

Richard Parris, Winstats(free from http://math.exeter.edu/rparris)

For the TI-83 Plus

APP - Probsim (free fromhttp://education.ti.com)

Age Number

0–4 38435–14 4226

15–24 19 975

25–44 27 20145–64 14 733

65–74 8 499

75+ 16 800

1

5

1 4 2 7 + =

5 10 10 10−

5 7 4 4 + =

10 10 10 5−

PROBABILITY




Probability

G3 demonstrate anunderstanding of thefundamental countingprinciple and apply it tocalculate probabilities ofdependent and independentevents

G3 How is the fundamental counting principle related to probability? Consider themarble situation described at the bottom of the previous page. The probability of

selecting red for the first marble is 35 , while the probability of selecting blue without

replacement is 2

4. The probability of selecting a red and a blue without replacement

would be ( ) 3 2 6 and

5 4 20P r b = × = .

Consider the experiment of a single toss of a standard die. There are six equally likelyoutcomes: 1, 2, 3, 4, 5, and 6. Define certain events as follows:

A: observe a 2B: observe a 6C: observe an even numberD: observe a number less than 5.

( ) 1

6P A = (observe a 2), ( ) 1

6P B = (observe a 6). What about P(A or B) (observe a

2 or 6)? This can be shown two ways:

( ) ( ) 1 1 2

total number of ways 6 6

n A n B+ += = or ( ) ( ) ( )

1 1 2 or

6 6 6P A B P A P B= + = + = .

Will this be true for any two events? The events “observe a 2,” and “observe a 6” arecalled mutually exclusive events, or disjoint because one can observe only a 2 or a 6,not both at the same time. On the other hand, events likeC and D above have at least one element in common, andtherefore are not mutually exclusive. Consider the eventsC and D.

The event (C or D) includes all the outcomes in C or Dor both. That is,

( or ) (observe an even number or a number less than five)

(observe 2, 4, 6, or observe 1, 2, 3, 4)

P C D P==

Every outcome except five is included in (C or D). Thus there are exactly 5 favourable

outcomes. Thus 5

( or )6

P C D = . But ( ) ( ) 3 4 7

6 6 6P C P D+ = + = , which cannot be

possible since it exceeds 1.

The outcomes 2 and 4 are contained in both C and D and are being counted twice.They must be removed. There is an overlap.

( ) ( ) ( ) ( )3 4 2 5

or and 6 6 6 6

P C D P C P D P C D= + - = + - =

PROBABILITY



Probability

Performance (G3)

1) Discuss whether the following pairs of events are mutually exclusive and whether they areindependent. [Note: some groups may be sensitive to examples using a standard deck ofcards.)a) The weather is fine; I walk to work.b) I cut a deck of cards obtaining a queen; you cut a 5.c) I cut the deck and have a red card; you cut a card with an odd number.d) I select a voter who registered Liberal; you select a voter who is registered Tory.e) I found a value for x to be greater than –2; you found x to have a value greater than

3.f) I selected two cards from the deck; the first was a face-card, the second was red.

2) If 366 different possible birthdays are each written on a different slip of paper and put ina hat and mixed,a) Find the probability of making one selection and getting a birthday in April or

October. [Ans: ]b) Find the probability of making one selection that is the first day of a month or a July

date. [Ans: ]

3) A store owner has three student part-time employees who are independent of each other.The store cannot open if all three are absent at the same time.a) If each of them averages an absenteeism rate of 5%, find the probability that the store

cannot open on a particular day. [Ans: ]

b) If the absenteeism rates are 2.5%, 3%, and 6% respectively for three differentemployees, find the probability that the store cannot open on a particular day.

c) Should the owner be concerned about opening in either situation a) or b)? Explain.

4) There are 6 defective bolts in a bin of 80 bolts. The entire bin is approved for shipping ifno defects show up when 3 are randomly selected.a) What is the probability of approval if the selected bolts are replaced? not replaced?b) Compare the results. Which procedure is more likely to reveal a defective bolt?

Which procedure do you think is better? Explain.

5) Mary randomly selects a marble from a bag containing 13 each of black, white, red andblue marbles. The marbles of each color are numbered from 1 to 13. What is theprobability that Mary will select either a 10 or a red? Below is Fred’s solution. Explainwhat Fred is thinking. Will his attempt lead to a correct answer? Explain.

Journal

6) Consider the table of experimental results. Comment on the following solution attempts.a) If one of the 2072 subjects is

randomly selected, the probability ofgetting someone who took Seldaneor a placebo is

b) If one of the 2072 subjects is randomly selected, the probability of getting someonewho took Seldane or experienced drowsiness can be found by


Computer Software:


For the TI-83 Plus


51366

12 31 1 42 7 +

366 366 366 366 61− = =

1 1 1 1 x x = 0.000125

20 20 20 8000=

781 665 14460.3489

2072 2072 4144+ = =

781 237 10180.491

2072 2072 2072+ = =

4 13 17 (10 or red)

52 52P

+= =

PROBABILITY




Probability

G6 demonstrate anunderstanding of thedifference between probabilityand odds

B7 calculate probabilities tosolve problems

G6 Expressions of likelihood are often given as odds. For example, 50:1, expressed“fifty to one,” is an expression of odds for a situation where the event is not verylikely to happen. The use of odds makes it easier to deal with money exchanges thatresult from gambling. The likelihood of an event can be expressed in terms of the

odds against that event, or the odds in favour. For example, if ( )25

P A = , then odds

against ( )

( )

335

2 25

P AA

P A= = = where A is the complement of A or “not A”. The answerer

is expressed as 3:2, or “three to two.” The corresponding odds in favour are 2:3.

In other words, odds against equals the ratio of unfavourable outcomes to favourableoutcomes. Odds in favour equals the ratio of favourable outcomes to unfavourableoutcomes.

For bets, the odds against an event represent the ratio of net profit to the amount bet.odds against event A = (net profit) : (amount bet)

Suppose a bet pays 50:1. If the odds aren’t specified as being in favour or against,they are probably the odds against the event occurring. If a person were to win a betwith 50:1 odds, that person would make a profit of $50 for each $1 bet. The personwould collect $51.

G6/B7 Suppose an electrical circuit has 50:1 odds against failure. What are the oddsagainst two such separate and independent circuits both failing? The best way tosolve this problem is to first convert the 50:1 odds to the corresponding probability

of failure 1

51

æ ö÷ç ÷ç ÷÷çè ø . Use the multiplication rule: 1 1 1

51 51 2601´ = . This gives the probability

of both circuits failing and is equivalent to odds of 2600:1.

Consider also a bag containing 50 black marbles and 1 white marble. The oddsagainst picking a white marble are 50:1 but the probability of selecting a white marbleis .

When performing calculations involving likelihoods, use probability values between 0and 1, not odds. This is why more time is spent on probability even though oddsseem to be heard more often.

1

51

PROBABILITY



Probability

Pencil and Paper (G6)

1) Ask students to complete the following conversions: [Note: Some groups may be sensitiveto situations involving gambling.]a) If P(A) = 2/7, find the odds against event A occurring. [Answer: 5:2]b) Find the probability of event A if the odds against it are 9:4. [Answer: 4/13]c) If the odds against an event are 7:3, what are the odds against

this event occurring in all of three separate and independent trials? [Answer: 973:27]d) In a fair game, all of the money lost by some players is won by

others. For one fair game, a $2 bet nets a profit of $16. Findthe odds against winning and find the probability of winning. [Answer: 8:1, 1/9]

e) A standard roulette wheel has 38 different slots numbered 1through 36 and 0, and 00. If you bet on any individual number,the casino gives you odds of 35:1. What would be fair odds ifthe casino did not have an advantage? [Answer: 37:1]

f) The actual odds against winning when you bet on “odds” atroulette are 10:9. What is the probability of winning? [Answer: 9/19]

Performance (G6/B7)

2) A slot machine has three drums, each of which contains different symbols, often fruit andbars. When the machine is activated all three drums roll. On this machine the first drumstops first, then the second drum, then the third. If the drums stop at a winningcombination of symbols, then coins will fall into a metal tray. The chance of thishappening depends on the distribution of the symbols on the drums. The table belowshows a typical distribution: [Note: Some groups may be sensitive to examples involvinggambling.]

a) Ask students to complete the following table:

b) How do you think the pay-off odds were determined?c) For each of the winning combinations above, calculate the average total dollars paid

out per 7500 plays.d) If these are the only winning combinations, about how many times, on average, can a

player expect to win per 7500 plays?

Nova Scotia. Department ofHealth, Drawing the Line: AResource for the Prevention ofProblem Gambling. Halifax:communications Nova Scotia,1997.


Computer Software:


For the TI-83 Plus


Symbol

cherry

lemon

plumorange

banana

bardouble bar

Drum 1

6

3

14

3

23

Drum 2

6

1

55

1

31

Drum 3

1

6

72

4

11

Winning Outcome

dbl bar, dbl bar, dbl barbar, bar, bar

plum, plum, bar

orange, orange, bananacherry, cherry, plum

lemon, lemon, lemon

cherry, cherry, lemonplum, lemon, cherry

Average Frequency Per

3 1 1 3´ ´ =

1 5 1 5´ ´ =

Probability

3/7500=

5/7500=

Payoff Odds

500 to 1

300 to 1400 to 1

20 to 1

3 to 175 to 1

5 to 1

2 to 1

PROBABILITY




Probability

G1 develop and applysimulations to solve problems

G1 Simulation is a procedure developed for answering questions about real problemsby running experiments that closely resemble the real situation.

Suppose the students want to find the probability that a family with 3 childrencontains exactly one girl. If students cannot compute the theoretical probability anddo not have the time to locate three-child families for observation, the best plan mightbe to simulate the outcomes for three-child families. One way to accomplish this is totoss three coins to represent the three births. A head could represent the birth of agirl. Then, observing exactly one head in a toss of three coins would be similar, interms of probability, to observing exactly one girl in a three-child family. Studentscould easily toss the three coins many times to estimate the probability of seeingexactly one head. The result gives them an estimate of the probability of seeing exactlyone girl in a three-child family. This is a simple problem to simulate, but the idea isvery useful in complex problems for which theoretical probabilities may be nearlyimpossible to obtain.

Students need work on connecting simulation results to the original problem. Whenchoosing a simple device to model the key components in the problem, they have tobe careful to choose a model that generates outcomes with probabilities to matchthose of the real situation. Students could use devices such as coins, dice, spinners,objects in a bag, random numbers tables and random number generators. On the TI-83 plus, the built-in function randBin (found under Math-PRB) can simulate manysituations.

Students need to understand that the experimental probability approaches thetheoretical probability as the number of trials increases. They should also realize thatknowing the probability of an event gives them no predicting power as to what theoutcome of the next trial will be. However, after enough trials, they should be able topredict with some confidence what the overall results will be.

When conducting simulations students should follow a process like the one outlinedbelow (see next page for an actual class activity):

Step 1: State the problem clearly. Step 2: Define the key components. Step 3: State the underlying assumptions. Step 4: Select a model to generate the outcomes for a key component. Step 5: Define and conduct a trial. Step 6: Record the observation of interest. Step 7: Repeat steps 5 and 6 until 50 trials are reached. Step 8: Summarize the information and draw conclusions.

PROBABILITY



Probability

Activity (G1)

1) Marie has not studied for her history exam. She knows none of the answers on theseven-question true-and-false section of the test. She decides to guess at all seven.Estimate the probability that Marie will guess the correct answers to four or moreof the seven questions. Ask students to complete the following:a) What is it that you are being asked to do?b) To perform a simulation, what assumptions should you make?c) Describe the model you would choose to perform the simulation.d) Pretend that you are watching the simulation. Describe what you observe for

the entire simulation.e) What conclusion do you think would be made?

2) Suppose a stick, or a piece of raw spaghetti, has been broken at two randompoints. What is the probability that the three pieces will form a triangle? (piecesmust touch end to end).a) Ask students to describe the process that might be used to estimate the answer

using experimental probability.b) Instead, ask students to conduct a simulation. Assume the spaghetti is 100 units

long. Generate two random numbers between 0 and 100 using each as a side ofa triangle. Determine the length of the third side. Check to see if the numbersrepresent the lengths of the side of a triangle?

c) Repeat a number of times to determine the experimental probability andcompare with others in your class.

Performance

3) Dale, a parachutist, jumps from an airplane and lands in a field. What are thechances that Dale will land in a particular numbered plot? Make a field grid using anormal sheet of graph paper divided into four equal areas.a) Model the situation by tossing a thumbtack onto the grid from a metre or more

away. (If the tack bounces off the sheet, don’t count it as a toss.) In yourresponse consider several questions:i) Is there an equal chance to land in each plot?ii) How many times did Dale land in plot 1?iii) Compare what was found in the experiment with what you expected to

find.

b) Conduct the experiment again, but use a field divided into plots A and B tofind the probability that Dale will land in Plot A.

c) Perform a simulation to answer the same problem as in(b). Compare the results of the simulation with that ofthe experiment. Comment.

4) Perform simulations to estimate the probability of eachevent.a) What is the probability that all five children in a family will be girls?


Computer Software:


For the TI-83 Plus


PROBABILITY




Probability

G4 apply area diagrams andtree diagrams to interpret anddetermine probabilities ofdependent and independentevents


G3 demonstrate anunderstanding of thefundamental countingprinciple and apply it tocalculate probabilities

G4 A certain restaurant offers select-your-own sandwiches. That is, a person mayselect one item from each of the categories listed. It is important to be sure that allthe possible outcomes are known and that they are equally likely. Only then can thetheoretical probability of events be calculated. A tree diagram is one way to do this:

From the tree diagram students can really see that there are 16 equally likely outcomesThey could use the Fundamental Counting Principle to check their results( )2 4 2 16´ ´ = .

G4/B7/G3 Using an area model gives a pictorial representation of the analysis, whichprovides visual insights into the concepts of probability. Reliance on geometric skillsallows the development of those concepts, which a lack of arithmetic skills wouldnormally impede. Dividing a region in proportion to the appropriate probabilitiesappeals to students’ intuitive understanding of probability. For example:

Rita has two dice, one red, one blue. Help her determine the probability of having thered die show an even number and the blue die an odd number. Using a square to

represent one, Rita thinks she should shade 1

2 of the square to represent the

probability, the red die will show an even number. Students shouldbe asked to explain why this makes sense. Have students completethe problem. They should then shade the upper half of the squareto represent the blue die showing an odd number (three oddnumbers of six possible numbers). The overlapped shaded regionwill indicate the probability of both events being true.

Using a similar method, ask students to find the probability that when Rita throwsboth dice, the red one shows a number less than five, and the blue one, a numbergreater than one.

The teacher might ask students to check their answers using tree diagrams and/or the

fundamental counting principle 4 5 20 5

6 6 36 9

æ ö÷ç ´ = = ÷ç ÷çè ø.

Blu

e di

e

Red die

odd

odd

even

even

Bread Filling Extras

whitewhole-wheat

tunachickencheeseegg

sproutslettuce

PROBABILITY



Probability

Performance (G4/B7/G3)

1) Barb and Ann are having a contest to see who can hit a target first. Both Barb andAnn have a 50% chance of hitting their target on each shot. If Barb lets Ann gofirst each time, what is the probability that Ann wins?

2) A certain restaurant offers select-your-own desserts. That is, a person may selectone item from each of the categories listed:

Newan, Claire et al. ExploringProbability. The QuantitativeLiteracy Series. Palo Alto,CA: Dale SeymourPublications, 1987.

a) Using a tree diagram, list all possible desserts that can be ordered.b) Would you expect the choices of a dessert to be equally likely for most

customers?c) If the probability of selecting chocolate mint ice cream is 40%, and vanilla ice

cream is 10%, chocolate sauce is 70%, and cherries 20%, describe the dessertwith the highest probability of being selected.

3) A certain model car can be ordered with one of three engine sizes, with or withoutair conditioning, and with automatic or manual transmission.a) Show, by means of a tree diagram, all the possible ways this model car can be

ordered.b) Suppose you want the car with the smallest engine, air conditioning, and

manual transmission. A car agency tells you there is only one of the cars onhand. What is the probability that it has the features you want, if you assumethe outcomes to be equally likely? [Ans: ]

4) In a restaurant there are four kinds of soup, 12 entrees, six desserts, and threedrinks. How many different four-course meals can a patron choose from? If 4 ofthe 12 entrees are chicken and two of the desserts involve cherries, what is theprobability that someone will order wonton soup, a chicken dinner, a cherry dessertand milk? [Ans: ]

5) Licence plates for cars often have three letters of the alphabet then three digits from0 to 9. How many possible different licence plates can be produced? What is theprobability of having the plate “CAR 000”?

6) A spinner is marked with an A or B as shown. Each round consists of either one ortwo spins. The player with the highest score wins. To begin the game, player 1spins first. If the spinner lands in the area marked A, player 1 scores a point, andthis ends the round, and player 2 spins again to begin round 2. If on the first spinthe spinner lands in the area marked B, then player 2 spins the spinner; player 2scores 2 points if the spinner lands in B, and player 1 scores 1 point if it lands in A.Use the square grid to.a) Find P(A will score on a given round).b) Find P(B will score on a given round).


Computer Software:


For the TI-83 Plus


Ice Cream Sauce Extras

vanillastrawberrychocolate mint

chocolatecaramel

cherriespeanuts

1

12

8 1864 and

864 108=

PROBABILITY




Probability

G7 distinguish betweensituations that involvepermutations andcombinations

G7 Before describing different situations in terms of permutations and combinations,students need to have an opportunity to solve simple counting problems (seeelaboration for G2/G3, p. 98). They may wish to organize their work into systematiclists and/or tree diagrams. As the number of choices increase, they should see the needfor a way to count more efficiently. For example:

a) How many different routes can you take from Sydney toHalifax through Antigonish?

b) How many routes are there from Antigonish to eitherHalifax or Sydney?

Following this, the class might be split into two groups—oneto do Problem A, the other Problem B. Students shouldpresent their solutions to the class.

Problem A: Suppose there were three people, Adam, Marie, and Brian, standing inline at a banking machine. In how many different ways could they order themselves?

Problem B: The executive of the student council has five members. In how manyways can a committee of three people be formed?

Solutions might look like the following:

Problem A: using a systematic list: A M B, A B M, M B A, M A B, B A M, B M A

Problem B: using a systematic list: if Adam, Marie, and Brian along with Dennis andElaine were on the executive, then to select committees of three, starting with Adam,Marie and Brian, the five permutations in the answer to A above would result in thesame five people being the committee, so they represent one combination. e.g., AMBon the committee is the same as MBA.

PROBABILITY



Probability

The essential difference between these two situations needs to be discussed andemphasized. Eventually, Problem A should be described as a permutation (order isimportant), Problem B as a combination (order not important).

Pencil and Paper (G7)

1) For each of the following, decide whether permutations or combinations areinvolved:a) the number of committees of 2 that can be formed from a group of 12 people

[Combination]b) the number of possible lineups for a baseball team that can be formed from 12

people (a baseball team consists of nine players, as follows: pitcher; catcher; first,second, and third basemen; shortstop; right, centre, and left-fielders)[Permutation]

c) the number of five-letter licence plates that can be formed from 12 differentletters [Permutation]

d) the number of six subsets that can be formed from 12 different letters[Combination]

e) the number of five-man basketball teams that can be formed from 10 players[Combination]

f) the number of ordered triples that can be formed from 10 different numbers[Permutation]

g) the number of ordered triples that can be formed from the numbers 1, 1, 1, 3,3, 5, 5, 5, 5, and 4 [Permutation]

2) The manager of a baseball team needs to decide the batting order for the seasonopener. In how many ways can the first four batters be arranged on the battingroster? Is this a permutation or combination question? Explain. [Ans: 24;Permutation]

3) As a promotion, a record store placed 12 tapes in one basket and 10 compact discsin another. Pierre was the one millionth customer and was allowed to select 4 tapesand 4 compact discs. To find how many selections that can Pierre make, does oneuse permutations or combinations? Explain. [Combination]

4) Three identical red balls (R) and two identical white balls (W) are placed in a box.How many ways are there of selecting the balls in the following order? [Ans: 12]

RWRRW

5) Find the total number of arrangements of the letters of the word “SILK.” [Ans: 24]


Computer Software:


For the TI-83 Plus


PROBABILITY




Probability

A6 develop an understandingof factorial notation and applyit to calculating permutationsand combinations

A6 As students refine their methods of counting, they are introduced ton! (n factorial) to represent the number of ways to arrange n distinct objects in a line.Example 1, the product rule can be used to find the number of possible arrangementsfor three people standing in a line. There are three people to choose from for the frontof the line. For each of these choices, there are two people to choose for the secondposition in the line. For each of these choices, there is one person to choose from theend of the line. Therefore, there are 6 possible arrangements.

Example 2, at a music festival, eight trumpet players competed in the Baroque class.After the judging, they were awarded 1st, 2nd, 3rd ... down to 8th place. In how manyways could their placements be awarded?

If all the trumpet players were given a standing, first, second, third, ... , eighth, thenthere are eight people eligible for first, which leaves seven eligible for second, sixpeople eligible for third ... leading to a calculation 8 7 6 5 4 3 2 1´ ´ ´ ´ ´ ´ ´ . Thisproduct can be written in a compact form as 8! and is read “eight factorial.”

In general, where and we define

A6 If there are only three prizes to be given, how many ways could placement beawarded?

Students should reason that eight people are eligible to come first, only seven areeligible to come second, and six are eligible to come third 8 7 6 336® ´ ´ ® . Thiscould be worded “How many permutations are there of eight distinct objects takenthree at a time?”

A1/G8 The symbol commonly used to represent this is 8P3, or nP3 for the number of“n” objects taken “r” at a time. Students should notice that

Students should note that when five people are tobe arranged in a straight line there would be 5! or120 ways to do this. However, if the same fivepeople were to be arranged around a table in theorder, say A, B, C, D, and E, their relativeposition to each other would not bedistinguishable.

Thus, the total number of arrangements would be: 5 5 5!

4! 245 5P

= = = .

( )

( )

8 3

8 3

8 3

8 3

8 7 6

8 7 6 5 4 3 2 1

5 4 3 2 1

8!

5!

8!

8 3 !

!In general,

- !n r

P

P

P

P

nP

n r

= ⋅ ⋅

⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅=

⋅ ⋅ ⋅ ⋅

=

=−

=

( )( ) ( )( )( )! 1 2 3 2 1 ,n n n n= - - ¼ n NÎ 0! 1.=

PROBABILITY



Probability

Pencil and Paper (A6)

1) The town of Karsville, population 32 505, is designing its own licence plates forresidents to place on the front of their automobiles.a) Ask students to use counting principles to determine the best of the following

three options and to explain their choice:

i) a licence made from using four single-digit numerals from 1 to 9ii) a licence made of three single-digit numerals from 1 to 9, and one letter

from the alphabetiii) a licence made from three single-digit numerals from 1 to 9, and two

letters from the alphabet.

b) Ask students to select the best combination of single-digits from 1 to 9 andletters to suit the purposes of this town, and defend their selection.

2) The figure shows three black marbles and two white marbles.Suppose they are in a box. Without looking in the box, randomlychoose two of the five marbles. How many ways are there to selecttwo marbles that are the same colour? Each a different colour?

Pencil and Paper (A6/G8)

6) a) Indicate which of the following are true (T) and which are false (F).

i)5!

5 44!

= ´ [ (F) ]

ii)10!

10 9 8 = 7!

´ ´ [ (T) ]

iii) 8P2 = 56 [ T ]iv) 100 4P = 100 99 98 97´ ´ ´ [ T ]

b) Create a story where each expression above would be used in the solution.

Paper and Pencil (A6/G8)

7) There are five non-collinear points on a plane.a) How many segments can be formed using these five points as endpoints?b) If consecutive points are joined, a convex polygon is formed. How many

diagonals does this polygon have?

8) A local pizza restaurant has a special on its 4-ingredient 20-cm pizza. If there are 15ingredients from which to choose, how many different “specials” are possible?


Computer Software:


For the TI-83 Plus


PROBABILITY




Probability

A6 develop an understandingof factorial notation and applyit to calculating permutationsand combinations

G7 distinguish betweensituations that involvepermutations andcombinations

A6/G7 There are five members on the executive of the students council. If these fivewere elected from a list of 10 candidates for executive positions, the number of ways10 people can be slotted into five positions would be found using permutations

( )10 5

10!30240

10 5 !P = =

- .

Now, from these five elected people a committee of three is struck: If the five peopleare represented by A, B, C, D, and E, then clearly a committee with A, B, and C isthe same as a committee with C, A, and B. So, the order of the selection is notimportant, and the arrangement is called a combination. Therefore, since ABC, ACB,BAC, BCA, CAB, and CBA are all considered the same committee, they representone combination. The number of permutations of A, B, and C is 3!. Thus, thenumber of committees from the original list of 10 candidates

10 310 3

number of ways the executive was chosen3!

30240

3!5040

That is 50403!

PC

=

=

=

= =

and the number of committees from the five people on the executive selected would

be 5 3

5 3 103!P

C = = .

Students should now apply combinations in a few simple problems where they would

be expected to use the formula ( )5 3

!! !

nr C

r n r= =

- or use technology that performs

combination calculations.

PROBABILITY



Probability

Performance (A6/G7)

1) Mrs. Sandhurst has the following books on her reading list: Great Expectations,Lord of the Flies, The Great Gatsby, Wuthering Heights, Fifth Business, The StoneAngel.

a) In how many ways can Mrs. Sandhurst arrange these books on her bookshelf?[Ans: 720]

b) What is the probability that Fifth Business is next to The Stone Angel on hershelf? [Ans: ]

c) If a student borrows two of the books before she arranges them on the shelf,how many fewer arrangements does she have? [Ans: 114]

d) If she arranges any four of the books on the shelf, how many fewer arrangementsdoes she have? [Ans: 96]

2) On the pinball machine below, a ball falls from the top to the bottom. How manydifferent paths can the ball follow assuming the ball falls without being pushedupwards? [Ans: 4 x 4 x 3 = 48]

3) In how many ways cana) a committee of four people be selected from eight people? [Ans: 70]b) a team of five players be selected from seven people? [Ans: 21]c) a study group of 4 people be selected from 10 students? [Ans: 210]

4) A local pizza restaurant has a special on its 5-ingredient 22-cm pizza. If there are 12ingredients from which to choose, how many different “specials” are possible?[Ans: 792]


Computer Software:


For the TI-83 Plus


13

Ball shot from starting position

PROBABILITY




Probability

B8 determine probabilitiesusing permutations andcombinations


B8/B7 Students should now apply nPr and nCr to probability problems.

One practical use of permutations and combinations is in the field of probability. Forexample, a deck of 52 cards is shuffled well. What is the probability that A, K, Q ofspades will be dealt to you as the first three cards?

Students might reason that since they want to see three particular cards from 52possible cards, they would use nPr or 52P3.

52 3

52! 52!132 600

(52 3)! 49!= Þ Þ

-P

and only one of those outcomes is favourable, so

1( , , )

132600=P A K Q

Combinations are sometimes used along with other counting techniques. Forexample, ask students to read the following problem, analyse Susan’s solution, identifyany error and report their findings:

Susan belongs to the school’s seven member in-line skaters club. The club has beenasked to select two girls and two boys to go to Toronto to take part in a skatersconvention. What is the probability that Susan will be selected if there are threeboys and four girls in the club?

Susan’s solution:– there are 4C2 ways to select two girls

– so, ( )4 2 4!/ 2!2! 6 waysC = =

– there are 3C2 ways to select two boys

– so, ( )3 2 3!/ 1!2! 3 waysC = =

– because there must be two girls and two boys, there are 6 + 3 = 9 ways offorming the group that is going

– if the four people are selected at random, the probability that Susan is selectedwould be 1 in 9

PROBABILITY



Probability

Performance (B8)

1) There are 30 students in your mathematics class. Three students are selected to siton a committee.a) How many committees can be formed if each member has equal status?b) How many committees can be formed if the first person chosen is the

chairman, the second is the secretary, and the third is the treasurer?

2) Five identical red balls (R) and two identical white balls (W) are placed in a box.How many distinguishable ways are there of selecting the balls in the followingorder?

RWRRWRR

Performance (B8/B7)

3) Nine people try out for nine positions on a baseball team. Each position is filled byselecting players at random. Assume all players are equally qualified for everyposition.a) In how many ways could the positions be filled?b) What is the probability that Duffy will be the pitcher?c) What is the probability that David, George or Duffy will be first baseman?d) What is the probability that David, George, or Duffy will be first baseman and

Eleanor or Georgina will be pitcher?

4) The numbers on a raffle ticket contain three digits. The first digit cannot be zero.a) What is the probability of ticket number 917 winning the grand prize? What

assumption did you make?b) What is the probability that a ticket with three as a second digit wins the grand

prize?

5) Three black marbles and two white marbles are in a box. Without looking in thebox player A randomly chooses two of the five marbles. If they are the same colour,player A wins, if they are a different colour, player B wins.a) What is the probability that player A wins? __________

What is the probability that player B wins? __________b) Some combinations of black and white marbles will produce a fair game. Can

you find a combination to make it a fair game? Can you find another?c) Create a simulation for this game.


Computer Software:


For the TI-83 Plus


PROBABILITY




Probability

Math 3206

Documents

Transcript of Math 3206