Mathematics - Memorial University of Newfoundlandcollections.mun.ca/PDFs/cmc_curr/Mathematics3204or3205.pdf · 2015-08-07 · Mathematics Mathematics 320413205 RJ.!I IJ.'II GOVERNMENT

Mathematics Mathematics 320413205

RJ.!I IJ.'II GOVERNMENT OF ri1aJ 1t.f1 NEWFOUNDLAND

AND LABRADOR 1!1 ~ Department of Education ~ ., Division of Program Development

A Curriculum Guide November 2002

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ACKNOWLEDGMENTS

Acknowledgments

MEMORIAL UNIVERSITY CURRICULUM MATERIALS

SEP - 3 2003

CENTRE OF NEWFOUNDLAND

The departments of education ofNew Brunswick, Newfoundland and

Labrador, Nova Scotia, and Prince Edward Island gratefully acknowledge

the contributions of the following groups and individuals toward the

development of this Mathematics 3204/3205 mathematics curriculum

guide.

The Regional Mathematics Curriculum Committee; current and past

representatives 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


Lynn Evans Phillips, Mathematics Teacher

Park View Education Centre

Newfoundland and Labrador

Sadie May, Distance Education Coordinator for Mathematics


Patricia Maxwell, Program Development Specialist


Prince Edward Island

Elaine Somerville, Mathematics/Science Consultant


The Provincial Curriculum Working Group, comprising teachers and

other educarors in Nova Scotia, which served as lead province in drafting

and revising the document.

The teachers and other educators and stakeholders across Atlantic

Canada who contributed to the development of the Mathematics 3204/

3205 mathematics curriculum guide.

ATLANTIC CANADA MATHEMATICS CURRICULUM· MATHEMATICS 3204/3205

Contents

Introduction

Curriculum Outcomes

Program Design and Components

Course Organization

CONTENTS

Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . .. . . ... . . . . . . . 1 Rationale .. .. ..... ... ... ...... ....... .. ....... ... ... .. .. .. ..... ..... ... .... ....... ... ......... .... .. 2 Meeting the Needs of All Learners ........ .... .. ... .... ..... .... .. .... ....... .... ..... . 4

Curriculum Outcomes Framework .... ..... .. ....... ..... .. .... .... ... ..... ... ..... .. . 5 Essential Graduation Learnings .. ... .... ............. .... .... ....... ..... .. ... ... .. .... .. 5 General Curriculum Outcomes ........ ........ ... .. .... ... ....... .. .. .. ... ..... ...... ... 6 Key-Stage Curriculum Outcomes .... ....... .. .. ... ..... .... .... .. ... ..... ...... .... ... 6 Specific Curriculum Outcomes ..... ....... .. .... ....... ... ........ .. .... ...... ... ... .... 6 SCO's for Mathematics 1204 and Mathematics 2204/2205 ...... .... ..... 7 SCO's and KSCO's for Mathematics 3204/3205 ... .............. ........ .. ... 17

Program Organization .. ... ............. ...... ... ........ ..... ........ ..... ...... ......... .. 29 Content Organization ... ... ........ ...... .... .... ... .. .... .. .. ....... ... .. ..... .. ......... . 31 Learning and Teaching Mathematics .... ..... .. ...... ... ....... .... ... .. .. ... .... .. .. 34 Summary of Changes in Instructional Practices .... ... .... ... .. .. ... ..... .. ... .. 35 Integrating Technology .. ... ....... ... ...... ................. ....... .. .. ...... .... ... .. ..... 35 Learning Resources ....... .. .. .... .......... ....... .... ...... ... .... ..... .. ..... ... ... .. ...... 36 Assessing and Evaluating Student Learning ........... ...... .... ... .......... .. ... 37

Course Design .... ........ ....... .... ..... ... ........... ..... ... ........... .... .. ... ..... .... ... 4 1 Unit 1: Quadratics .... ........ ... ......... .... ..... ....... .. .. .... ... ..... ... ..... ....... .. .. 43 Unit 2: Rate of Change ..... ... ........ ... ........... ... .... .... ..... ...... .............. .. 69 Unit 3: Exponential Growth .. ......... ... .. .. .. .. .... .. .. ..... ....... ..... ....... ....... 83 Unit 4: Circle Geometry ..... ... ..... .. ........ .. .. ............... ..... ..... .. ........ .. . 115 Unit 5: Probability ........... ... ... ...... ............. ....... .. .. .. ... .. ................... 143

ATLANTIC CANADA MATHEMATICS CURRICULUM MATHEMATICS 3204/3205 iii

Introduction

Background

INTRODUCTION

The mathematics curriculum for Atlantic Canada has been written in an

effort to align the outcomes for student learning in mathematics with the

recommendations of Curriculum and Evaluation Standards for School

Mathematics (National Council ofTeachers ofMathematics, 1989). This

document identifies the primary goal for all students to be the

attainment of mathematical power-the ability to make mathematical

connections, to reason logically, to communicate and apply mathematics

effectively in problem situations. Since the late 1980s, several influential

publications have affirmed this goal. In addition to Curriculum and

Evaluation Standards for School Mathematics, these include two

publications from the Mathematical Sciences Education Board

Everybody Counts: A Report to the Nation on the Future of Mathematics

Education (1989) and Reshaping Schoof Mathematics (1990). As well, the

National Council ofTeachers of Mathematics (NCTM) published the

companion standards documents Professional Standards for Teaching

Mathematics in 1991 and Assessment Standards for Schoof Mathematics in 1995. In April2000, the NCTM published its newest document, Principles and Standards for School Mathematics, a revision, rewriting, and

restatement of the 1989 Curriculum and Evaluation Standards for Schoof

Mathematics.

Foundation for the Atlantic Canada Mathematics Curriculum (1996)

firmly established Curriculum and Evaluation Standards for School

Mathematics (NCTM, 1989) as a guiding beacon for pursuing this vision, a vision that fosters the development of mathematically literate students. Curriculum design has been motivated by a desire to ensure that students benefit from world-class curriculum and instruction in

mathematics as a significant part of their school learning experience. More and more, students are being challenged to become problem solvers, to understand mathematical concepts by becoming active learners in highly interactive learning experiences. Computers and calculators are

becoming common classroom tools, and innovations in the assessment of student learning (which include portfolios and open-ended questions) are being used in classrooms.

Mathematics curriculum development in this region has taken place under the auspices of the Atlantic Provinces Education Foundation

(APEF), an organization sponsored and managed by the governments of the four Atlantic Provinces. The development process has brought

together teachers and Department of Education officials to co

operatively plan and execute the development of curricula in mathematics, science, language arts, and some other subject areas. Each

ATLANTIC CANADA MATHEMATICS CURRICULUM MATHEMATICS 3204/3205

INTRODUCTION

Rationale

2

of these curriculum efforts has been aimed at producing a program that

would ultimately support the essential graduation learnings (EGLs), also

developed regionally. The essential graduation learnings, and the

contribution of the mathematics curriculum to their achievement, are

presented in the Outcomes section of the Foundation for the Atlantic

Canada Mathematics Curriculum.

Foundation for the Atlantic Canada Mathematics Curriculum provides an

overview of the philosophy and goals of the public school mathematics

curriculum, presenting broad curriculum outcomes and addressing a

variety of issues with respect to the learning and teaching of

mathematics. It describes the mathematics curriculum in terms of a

framework of outcomes-general curriculum outcomes (GCOs), which

relate to subject strands, and key-stage curriculum outcomes (KSCOs),

which identify what students are expected 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 foundation

document by relating specific curriculum outcomes (SCOs) to each

KSCO and providing suggestions for learning experiences, instruction,

assessment, and resources.

The purposes of high school mathematics are embedded in a context that

is broad and consistent with accelerating changes in today's society-a

society that is increasingly dominated by technology and quantitative

methods. Predictions are that high school graduates in the future will

change careers at least four or five times. If we are to develop curriculum

for students who need to be flexible with respect to the workplace and

capable oflifelong learning, high school mathematics must emphasize a

dynamic form of literacy, and high school mathematics instruction must

maximize the opportunity for students to achieve outcomes dealing with

a broad range of topics. Experiences must be provided that encourage

and 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 the

need for all students to understand the complexities and technologies of

communication, to ask questions, to assimilate unfamiliar information,

and to work co-operatively. These needs are best addressed by developing

a curriculum that reflects the following beliefs:

• Knowing mathematics is ccdoing mathematics. "

Mathematics is more than just a collection of concepts and skills to be mastered; it includes methods of investigating and reasoning, means of communication, and notions of context. Instructional


INTRODUCTI ON

settings and student activities should be developed and grow out of problem situations. This view oflearning is summarized in Everybody Counts (Mathematical Sciences Education Board, 1989) .

"In reality, no one can teach mathematics. Effective teachers are those who can stimulate students to learn mathematics. Educational research offers compelling evidence that students learn mathematics when they construct their own mathematical understanding. To understand what they learn, they must enact for themselves verbs that permeate the mathematics curriculum: 'examine, represent, transform, solve, apply, prove, communicate' . This happens most readily when they are in groups, engage in discussion, make presentations, and in other 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 all intellectual disciplines, and this phenomenon has changed the fundamental mathematical ideas needed. Although traditional topics remain very important components of the curriculum, there is a shift in emphasis from a curriculum dominated by memorization of isolated facts and procedures and by proficiency with paper-and-pencil skills to one that emphasizes conceptual understanding, multiple representations and connections, mathematical modelling, and problem solving.

The integration of ideas from algebra and geometry is particularly strong, with graphical representation playing an important connecting role. Frequent references to graphing utilities indicate the value of having computers with appropriate graphing software and/or graphing calculator availability for students. Topics from statistics and probability are now elevated to a more central position for all students.

Arithmetic computation is not a direct object of study in the high school mathematics curriculum; however, conceptual and procedural understandings of number, numeration, and operations and the ability to make estimations and approximations to judge the reasonableness of results are strengthened in the context of applications and problem solving. Emphasis is placed on the role of technology and appropriate concepts and skills related to their use.

Changes in technology and the broadening of the areas in which mathematics is applied have resulted in growth and changes in the discipline of mathematics itself.

New technology not only has made calculations and graphing easier, it has changed the very nature of the problems important to mathematics and the methods m athematicians use to investigate them. Because technology is changing mathematics and its uses, students should learn to use graphing calculators and computers as

ATLANTIC CANADA MATHE MATICS CUR RICULUM: MATHEMATI CS 3204/3205 3

INTRODUCTION

Meeting the Needs of All Learners

4

tools for processing information and performing calculations to investigate and solve problems.

The visualization approach offered through the use of graphing utilities such as the graphing calculator affords more students greater access to more mathematics. With the wide availability of technology comes additional decision making regarding what skills need to be developed mentally. Some aspects of further development in mathematics are facilitated when students reach an automatic response level with respect to certain basic skills.

An important emphasis in this curriculum is the need to deal successfully

with a wide variety of equity and diversity issues. Not only must

teachers be aware of, and adapt instruction to account for, differences in

student readiness as they begin Mathematics 320413205 and as they

progress, but they must also remain aware of the importance of avoiding

gender and cultural biases in their teaching. Ideally, every student should

find his/her learning opportunities maximized in the mathematics

classroom.

The reality of individual student differences must be recognized as

teachers make instructional decisions. While this guide for Mathematics 320413205 presents specific curriculum outcomes for the course, it must

be acknowledged that all students will not progress at the same pace and

will not be equally positioned with respect to attaining a given outcome

at any given time. The specific curriculum outcomes represent, at best, a

reasonable framework for helping students to ultimately achieve the key

stage and general curriculum outcomes.


INTRODUCTION

Curriculum Outcomes Curriculum Outcomes Framework

Essential Graduation Learnings

The mathematics curriculum is based on a framework of outcomes

statements articulating what students are expected to know, be able to

do, and value as a result of their learning experiences in mathematics.

This framework comprises statements of the essential graduation

learnings, general curriculum outcomes, key-stage curriculum outcomes,

and specific curriculum outcomes. Foundation for the Atlantic Canada

Mathematics Curriculum articulates general curriculum outcomes and

key-stage curriculum outcomes. Curriculum guides provide specific

curriculum outcomes for each course, together with elaborations and

suggestions for related instructional and assessment strategies and tasks.

Teachers and administrators are expected to refer to the curriculum

outcomes framework to design learning environments and experiences

that reflect the needs and interests of the students.

Essential graduation learnings are statements describing the knowledge,

skills, and attitudes expected of all students who graduate from high

school. Essential graduation learnings are cross-curricular in nature and

comprise different areas oflearning: aesthetic expression, citizenship,

communication, personal development, problem solving, technological

competence and spiritual and moral development.

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

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

Communication: Graduates will be able to use the listening, viewing, speaking, reading, and writing modes oflanguage(s) and mathematical and scientific concepts and symbols to think, learn, and communicate effectively.

Personal Development: Graduates will be able to continue to learn and to pursue an active, healthy lifestyle.

Problem Solving: Graduates will be able to use the strategies and processes needed to solve a wide variety ofproblems, including those requiring language and mathematical and scientific concepts.

Technological Competence: Graduates will be able to use a variety of technologies, demonstrate an understanding of technological applications, and apply appropriate technologies for solving problems.

Spiritual and Moral Development: Graduates will demomtrate an understanding and appreciation for the place of beliefsystem in shaping the development of moral values and ethical conduct.

See Foundation for the Atlantic Canada Mathematics Curriculum, pages

4-6.

ATLANTIC CANADA MATHEMATICS CURRICULUM: MATHEMATICS 3204/3205 5

L

INTRODUCTION

General Curriculum Outcomes

Key-Stage Curriculum Outcomes

Specific Curriculum Outcomes

6

General curriculum outcomes are statements that identifY what students

are expected to know and be able to do upon completion of study in

mathematics. General curriculum outcomes contribute to the attainment

of the essential graduation learnings and are connected to key-stage

curriculum outcomes. The seven general curriculum outcomes for

mathematics are organized in terms of four content strands: number

concepts/number and relationship operations; patterns and relations;

shape and space; and data management and probability.

Number C oncepts/Number and Relationship O perations

• Students will demonstrate number sense and apply number theory concepts. Students will demonstrate operation sense and apply operation principles and procedures in both numeric and algebraic situations.

Patterns and Relations

• Students will explore, recognize, represent and apply patterns and relationships, both informally and formally.

Shape and Space

Students will demonstrate an understanding of and apply concepts and 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 and analysis of data.

• Students will represent and solve problems involving uncertainty.

Key-stage curriculum outcomes (KSCOs) are statements that identifY

what students are expected to know and be able to do by the end of

grades 3, 6, 9, and 12 as a result of their cumulative learning experiences

in mathematics. This curriculum guide lists key-stage curriculum

outcomes for the end of grade 12 (see pp. 17-26). Specific curriculum

outcomes for Mathematics 3204/3205 are referenced to key-stage

curriculum outcomes on these same pages.

Specific curriculum outcomes, which contribute to the achievement of

the key-stage curriculum outcomes, are statements identifYing what

students are expected to know and be able to do at a particular grade

level.

In the table that follows, the specific outcomes for Mathematics 1204 and Mathematics 220412205 are listed.

ATLANTIC CANADA MATHEMATICS CURRICULUM: MATHEMATICS 3204/3205

INTRODUCTION

SCO's for Mathematics 1204 and Mathematics 2204/2205 GCO A: Students will demonstrate number sense and apply number theory concepts.

Elaboration: Number sense includes understanding number meanings, developing multiple relationships among

numbers, 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 Mathematics 1204 and Mathematics 2204/2205

By the end of M athematics 1204, students will be

expected to

A 1 relate sets of numbers to solutions of inequalities

A1 analyse graphs or charts of situations to identifY

specific information

A3 demonstrate an understanding of the role of

irrational numbers in applications

A4 approximate square roots

AS demonstrate an understanding of the zero product

property and its relationship to solving equations by

facto ring

A6 apply properties o f numbers when operating upon

expressions and equations

A7 demonstrate an understanding of and apply the

proper use of discrete and continuous number systems

AS d emonstrate an understanding of and apply

properties to operations involving square roots

By the end of Mathematics 220412205, students will be

expected to

AI demonstrate an understanding of irrational

numbers in applications

A3 demonstrate an understanding of the application

of random numbers to statistical sampling

A4 demonstrate an understanding of the conditions

under which matrices have identities and inverses

AS demonstrate an understanding of the properties of

matrices and apply them

ATLANTIC CANADA MATHEMATICS CU RRICULUM : MATHEMATICS 3204/3205 7

INTRODUCTION

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, and

acquiring insights into the effects of an operation on a pair of numbers. Operation principles and procedures

would include such items as the effect of identity elements, computational strategies, and mental mathematics.




expected to

Bl model (with concrete materials and pictorial

representations) and express the relationships between

arithmetic operations and operations on algebraic

expressions and equations

B2 develop algorithms and perform operations on

irrational numbers

B3 use concrete materials, pictorial representations,

and symbolism to perform operations on polynomials

B4 identifY and calculate the maximum and/or

minimum values in a linear programming model

BS develop, analyse, and apply procedures for matrix

multiplication

B6 solve network problems using matrices


expected to

BI demonstrate an understanding of the relationship

between operations on fractions and rational algebraic

expressiOns

B2 demonstrate an understanding of the relationship

between operations on algebraic and matrix equations

B4 use the calculator correctly and efficiently

BS analyse and apply the graphs of the sine and cosine

functions

B6 derive and analyze the Law of Sines, Law of

Cosines, and the formula "area of a triangle

ABC= 2_bcsinA " 2

Bll develop and apply the procedure to obtain the

inverse of a matrix

BI2(Adv) derive and apply the procedure to obtain

the inverse of a matrix

Bl3 solve systems of equations using inverse matrices

BI4(Adv) determine the equation of a plane given

three points on the plane

BIS solve systems of"m" equations in " n" variables

with and without technology

8 ATLANTIC CANADA MATHEMATICS CURRICULUM MATHEMATICS 3204/3205

I

,

INTRODUCTION

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

concrete materials to polynomial and exponential functions. The representation of patterns and

relationships will take on multiple forms, including sequences, tables, graphs, and equations, and these

representations will be applied as appropriate in a wide variety of relevant situations.


for Mathematics 1204 and Mathematics 220412205


expected to

C1 express problems in terms of equations and vice

versa

C2 model real-world phenomena with linear,

quadratic, exponential, and power equations and

linear inequalities

C3 gather data, plot the data using appropriate scales

and demonstrate an understanding of independent

and dependent variables, and domain and range

C4 create and analyse scatter plots using appropriate

technology

C5 sketch graphs from words, tables, and collected

data

C6 apply linear programming to find optimal

solutions to real-world problems

C7 model real-world situations with networks

CB identifY, generalize, and apply patterns

C9 construct and analyse graphs and tables relating

two variables

C10 describe real-world relationships depicted by

graphs, tables of values, and written descriptions

C11 write an inequality to describe its graph

C12 express and interpret constraints using

inequalities

Cl3 determine and interpret the slope andy-intercept

of a line from a table of values or a graph

C14 determine the equation of a line using the slope

andy-intercept

C15 develop and apply strategies for solving problems


expected to

CI model situations with sinusoidal functions

C2 create and analyse scatter plots of periodic data

C3 determine the equations of sinusoidal functions

C4(Adv) determine the equations of sinusoidal

functions expressed in radians

C5 determine quadratic functions using systems of

equations

CB demonstrate an understanding of real-world

relationships by translating between graphs, tables ,

and written descriptions

C9 analyse tables and graphs of various sine and

cosine functions to find patterns, identifY characteristics, and determine equations

C10(Adv) analyse tables and graphs of various sine

and cosine functions to find patterns, identifY

characteristics, and determine equations using radians

C12 interpret geometrically the relationships between

equations in systems

C13 demonstrate an understanding that an equation

in three variables describes a plane

C 14 demonstrate an understanding of the

relationships between equivalent systems of equations

C 15 demonstrate an understanding of sine and cosine

ratios and functions for non-acute angles

ATLANTIC CANADA MATHEMATICS CURRICULUM MATHEMATICS 3204/3205 9

INTRODUCTION

GCO C: Students will explore, recogmze, represent, and apply patterns and relationships, both informally and formally.

Elaboration: Patterns and relationships run the gamut from number patterns and those made from concrete

materials to polynomial and exponential functions. The representation of patterns and relationships will take

on multiple forms, including sequences, tables, graphs, and equations, and these representations will be applied

as appropriate in a wide variety of relevant situations.


for M athematics 1204 and Mathematics 220412205


expected to

Cl6 interpret solutions to equations based on context

Cl7 solve problems using graphing technology

Cl8 investigate and find the solution to a problem by

graphing two linear equations with and without

technology

C19 solve systems of linear equations using substitution and graphing methods

C20 evaluate and inrerpret non-linear equations using

graphing technology

C21 explore and apply functional relationships

informally

C22 analyse and describe transformations of quadratic

functions and apply them to absolute value functions

C23 express transformations algebraically and with

mapping rules

C24 rearrange equations

C25 solve equations using graphs

C26 solve quadratic equations by factoring

C27 solve linear and simple radical, exponenrial, and

absolute value equations and linear inequalities

C28 explore and describe the dynamics of change

depicted in tables and graphs

C29 investigate and make and test conjectures

concerning the steepness and direction of a line

C30 compare regression models oflinear and nonlinear functions

C31 graph equations and inequalities and analyse

graphs, both with and without graphing technology


expected to

C 16(Adv) demonstrate an understanding of sine and

cosine ratios and functions for non-acute angles

expressed in radians

C17(Adv) solve problems by determining the

equation for the curve of best fir using sinusoidal regress JOn

Cl8 interpolate and extrapolate to solve problems

C19 solve problems involving systems of equations

C21 describe how various changes in the parameters

of sinusoidal equations affect their graphs

C22(Adv) describe how various changes in the parameters of sinusoidal equations, expressed in

radians, affect their graphs

C23 identifY periodic relations and describe their characteristics

C24 derive and apply the reciprocal and Pythagorean

identities

C25 prove trigonometric identities

C27 apply function notation to trigonometric

equations

C28 analyse and solve trigonometric equations with

and without technology

10 ATLANTIC CANADA MATHEMATICS CURRICULUM. MATHEMATICS3204/3205

INTRODUCTION

GCO C: Students will explore, recogmze, represen t, and apply patterns and relationsh ips, both informally and formally.


materials to polynomial and exponential functions. T he representation of patterns and relationships will take






expected to

C32 determine if a graph is linear by plotting points

in a given situation

C33 graph by constructing a table of values, by using

graphing technology, and, when appropriate, by the

slope y- intercept method

C34 investigate and make and test conjectures about

the solution to equations and inequalities using

graphing technology

C35 expand and factor polynomial expressions using

perimeter and area models

C 36 explore, determine, and apply relationships

between perimeter and area, surface area, and volume

C37 represent network problems using matrices and

vice versa


expected to

C 29 (Adv) analyse and solve trigonometric equations

wi th and without technology, expressing solutions in

radians

C 3 0 demonstrate an understanding of the relationship

between solving algebraic and trigonometric equations

ATLANTIC CANADA MATHEMATICS CURRICU LU M · MATHEMATICS 3204/3205 ,,

INTRODUCTION

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, using

appropriate measurement units and using formulas (e.g., surface area, Pythagorean Theorem) and/or procedures

(e.g., proportions) to determine measurements indirectly.


for Mathematics 1204 and Mathematics 2204/2205.


expected to

D1 determine and apply formulas for perimeter, area,

surface area, and volume

D2 apply the properties of similar triangles

D3 relate the trigonometric functions to the ratios in

similar right triangles

D4 use calculators to find trigonometric values of

angles and angles when trigonometric values are

known

D5 apply trigonometric functions to solve problems

involving right triangles, including the use of angles of

elevation of and apply ratios within similar triangles

D6 solve problems involving measurement using

bearings and vectors

D7 determine the accuracy and precision of a

measurement

DB solve problems involving similar triangles and

right triangles

D9 determine whether differences in repeated

measurements are significant or accidental

D 10 determine and apply relationships between the

perimeters and areas of similar figures , and between

the surface areas and volumes of similar solids

D 11 explore, discover, and apply properties of

maximum area and volume

D12 solve problems using the trigonometric ratios

D 13 demonstrate an understanding of the concepts of

surface area and volume

D 14 apply the Pythagorean Theorem


expected to

D1 derive, analyse, and apply angle and arc-length

relationships

D2 demonstrate an understanding of the connection

between degree and radian measure and apply them

D3 apply sine and cosine ratios and functions to

situations involving non-acute angles

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

the formula "area of a triangle ABC= .!_be sin A" to 2

solve problems


INTRODUCTION

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 is

characterized by such geometric relationships as (i) the direction, orientation, and perspectives of objects in

space, (ii) the relative shapes and sizes of figures and objects, and (iii) how a change in shape relates to a change

in size. Geometric concepts, properties, and relationships are illustrated by such examples as the concept of

area, the property that a square maximizes area for rectangles of a given perimeter, and the relationships among

angles formed by a transversal intersecting parallel lines .



By the end of Mathematics 1204, students wilL be

expected to

El explore properties of, and make and test

conjectures about, two- and three-dimensional figures

E2 solve problems involving polygons and polyhedra

E3 construct and apply altitudes, medians, angle

bisectors, and perpendicular bisectors to examine

their intersection points

E4 apply transformations when solving problems

E5 use transformations to draw graphs

E6 represent network problems as digraphs

E7 demonstrate an understanding of and write a proof for the Pythagorean Theorem

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

conjectures

E9 use deductive reasoning and construct logical arguments and be able to determine, when given a logical argument, if it is valid

Ell draw nets of various polyhedra


expected to

El demonstrate an understanding of the position of

axes in 3-space

E2 locate and identify points and planes in 3-space

ATLANTIC CANADA MATHEMATICS CURRICULUM. MATHEMATICS 3204/3205 13

INTRODUCTION

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 and

issues, (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.



By the end of Mathematics 1204, students wiLL be

expected to

F 1 design and conduct experimen rs using statistical methods and scientific inquiry

F2 demonstrate an understanding of concerns and issues that perrain to the collection of data

F3 construct various displays of data

F4 calculate various statistics using appropriate technology, analyse and interpret displays, and describe the relationships

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

F6 solve problems by modelling real-world phenomena

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

F8 determine and apply a line of best fir using the least squares method and median-median method, with and without technology, and describe the differences between the two methods

F9 demonstrate an intuitive understanding of correlation

FlO use interpolation, extrapolation, and equations to

predict and solve problems

Fll describe real-world relationships depicted by graphs and tables of values

Fl2 explore measurement issues using the normal curve

F13 calculate and apply mean and standard deviation using technology to determine if a variation makes a difference

Fl4 make and interpret frequency bar graphs while conducting experiments and exploring measurement ISSUes


expected to

Fl draw inferences about a population from a sample

F2 identify bias in data collection, interpretation and presentation

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

F6(Adv) explore periodic data to determine rhe equations of sinusoidal curves using regression analysis

F7 draw inferences from graphs, rabies, and reports

F8 apply characteristics of normal distributions

F9 construct, interpret, and apply 90% box plots

FlO interpret and apply histograms and probabiliry bar graphs

Fll determine, interpret, and apply confidence intervals

FI4 formulate hypotheses and null hypotheses

F15 design and conduct experiments/surveys to explore sampling variability

F16 demonstrate an understanding that the type of experiment/survey affects the organization and communication of results

F18 test hypotheses and interpret the results

Fl9(Adv) apply and interpret the chi-square ( z 2)

statistic

F20(Adv) collect data about two populations and analyse ir using rhe chi-square statistic


,

I

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INTRODUCTION

GCO G: Students will represent and solve problems involving uncertainty.

Elaboration: Representing and solving problems involving uncertainty entails (i) determining probabilities by

conducting experiments and/or making theoretical calculations, (ii) designing simulations to determine probabilities in situations that do not lend themselves to direct experiment, and (iii) analysing problem

situations to decide how best to determine probabilities.




expected to


expected to

G 1 construct and apply 90% box plots and normal

probability distributions and determine confidence

intervals

G2(Adv) connect probability with the chi-square

( z2) statistic to interpret its meaning

G3 graph sample distributions and interpret them using 90% box plots, probability bar graphs, and the language of probability


INTRODUCTION

Independent Study

16

L

Further to General Curriculum Outcomes, an Independent Study unit

has been developed for Mathematis 2204/2205. The purpose is to

prepare students for independent learning by providing students with an

opportunity to research some new mathematical topics and present their

new mathematical understandings to other students.

After participating in this unit of work, students will be exprected to

11 demonstrate an understanding of a mathematical topic through independent research

12 communicate the results of the independent research

I3 demonstrate an understanding of the mathematical topics presented by other students

ATLANTI C CANADA MATHEMATICS CURRICULU M MATHEMATICS 3204/3205

Key-Stage Curriculum Outcomes

Specific Curriculum Outcomes

INTRODUCTION

Key-stage curriculum outcomes (KSCOs) are statements that identify

what students are expected to know and be able to do by the end of grades 3, 6, 9, and 12 as a result of their cumulative learning experiences

in mathematics. This curriculum guide lists key-stage curriculum

outcomes for the end of grade 12 (see pp. 17-26). Specific curriculum

outcomes for Mathematics 3204/3205 are referenced to key-stage

curriculum outcomes on these same pages.

Specific curriculum outcomes are statements identifying what students

are expected to know and be able to do at a particular grade level, which

contribute to the achievement of the key-stage curriculum outcomes.

The numbers that follow each outcome statement are the page numbers

in this document where that outcome is being addressed. It should be noted the same outcomes are addressed a number of times.

In the tables that follow, the Mathematics 3204/3205 specific curriculum

outcomes are related to the key-stage outcomes.

The outcomes which apply only the Mathematics 3205 are indicated

with (Adv). The rest of the outcomes apply to Mathematics 3204 and

3205 .


INTRODUCTION

SCOs and KSCOs for Mathematics 3204/3205

GCO A: Students will demonstrate number sense and apply number theory concepts.

Elaboration: Number sense includes understanding number meanings, developing multiple relationships among

numbers, 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 the

outcomes for entry-grade 9 and will also be expected to

KSCO i: demonstrate an understanding of number

meanings with respect to the real numbers

KSCO ii: order real numbers, represent them in

multiple ways (including scientific notation), and

apply appropriate representations to solve problems

KSCO iii: demonstrate an understanding of the real

number system and its subsystems by applying a

variety of number theory concepts in relevant

Situations

KSCO iv: Some post-secondary intending students

will be expected to explain and apply relationships

among real and complex numbers

Specific C urriculum O utcomes (SCO )

By the end of Mathematics 3204/3205, students will be

expected to

A3 demonstrate an understanding of the role of

irrational numbers in applications (66)

A4 demonstrate an understanding of the nature of the

roots of quadratic equations (66)

A5 demonstrate an understanding of the role of real

numbers in exponential and logarithmic expressions

and equations (96)

A6 develop an understanding of factorial notation and

apply it to calculating permutations and combinations

(158, 160)

A7 describe and interpret domains and ranges using

set notation (50, 58, 88, 102, 11 0)

A9 represent non-real roots of quadratic equations as

complex numbers (66)


I

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INTRODUCTION

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, and acquiring insights into the effects of an operation on a pair of numbers. Operation principles and procedures

would include such items as the effect of identity elements, computational strategies, and mental mathematics.




KSCO i: explain how algebraic and arithmetic

operations are related, use them in problem-solving situations, and explain and demonstrate the power of

mathematical symbolism

KSCO ii: derive, analyse, and apply computational procedures (algorithms) in situations involving all

representations of real numbers

KSCO iii: derive, analyse, and apply algebraic

procedures (including those involving algebraic expressions and matrices) in problem situations

KSCO iv: apply estimation techniques to predict and justifY the reasonableness of results in relevant problem situations involving real numbers

KSCO v: Some post-secondary-intending students will be expected to apply operations on complex numbers to solve problems

Specific Curriculum Outcomes (SCO)


expected to

B 1 demonstrate an understanding of the relationships

that exist between arithmetic operations and the operations used when solving equations (60, 62, 64,

106, 108)

B2 demonstrate an understanding of the recursive nature of exponential growth (84, 86)

B4 calculate average rates of change (70, 72, 74, 76)

B8 determine probabilities using permutations and combinations (164, 166, 168)

B10 derive and apply the quadratic formula (62)

B11(Adv) analyse the quadratic formula to connect its

components to the graphs of quadratic functions (64)

B12 apply real number exponents in expressions and

equations (1 06,1 08)

B 13 demonstrate an understanding of the properties

oflogarithms and apply them (112)


INTRODUCTION





as appropriate in a wide variery of relevanr situations.




KSCO i: model real-world problems using functions,

equations, inequalities, and discrete structures

KSCO ii: represent functional relationships in multiple ways (e.g., written descriptions, tables,

equations, and graphs) and describe connections

among these representations



expected to

Cl model real-world phenomena using quadratic

functions (48, 50, 52, 54)

C2 model real-world phenomena using exponential functions (84, 86, 92, 94, 98)

C3 sketch graphs from descriptions, tables, and collected data (44, 50, 52, 98)

C4 demonstrate an understanding of patterns that are arithmetic, power, and geometric and relate them to

corresponding functions (44, 46, 86)

C8 describe and translate between graphical, tabular, written, and symbolic representations of quadratic relationships (48, 50, 52)

C9 translate between different forms of quadratic

equations (58, 60)

ClO(Adv) determine the equation of a quadratic

function using finite differences (56)

Cll describe and translate between graphical, tabular, written, and symbolic representations of exponential

and logarithmic relationships (86, 92, 94, 110)

20 ATLANTIC CANADA MATHEMATICS CURRICULUM· MATHEMATICS 3204/3205

INTRODUCTION









KSCO iii: interpret algebraic equations and

inequalities geometrically and geometric relationships

algebraically

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


By the end of Mathematics 3204/3205, students will be

expected to

CIS relate the nature of rhe roots of quadratic

equations and the x-intercepts of the graphs of the

corresponding functions ( 66)

CI6 demonstrate an understanding that slope depicts

rate of change (72, 7 4)

CI7 demonstrate an understanding of the concept of

rate of change in a variety of situations (70, 72, 74, 76, 78, 80)

CIS demonstrate an understanding that the slope of a line tangent to a curve is the instantaneous rate of

change of the curve at a point of tangency (78, 80)

C I9 demonstrate an understanding, algebraically and

graphically, that the inverse of an exponential function

is a logarithmic function (11 0)

C20(Adv) represent circles using parametric equations (138)

C22 solve quadratic equations (62, 64)

C23 solve problems involving quadratic equations (52,60)

C24 solve exponential and logarithmic equations (106, 108, 112)

C25 solve problems involving exponential and

logarithmic equations (86, 88, 90, 92, 98, 106, 1 08)

C27 approximate and interpret slopes of tangents to

curves at various points on the curves, with and without technology (78, 80)

C28 solve problems involving instantaneous rare of change (78)


INTRODUCTION

GCO C: Students will explore, recogmze, represent, and apply patterns and relationships, both informally and formally.

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


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

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

KSCO vi: perform operations on and between functions

KSCO vii: Some post-secondary-intending students will be expected to describe and explore the concept of continuity of a function

KSCO viii: Some post-secondary-imending studems will be expected to investigate limiting processes by examining infinite sequences and series

KSCO ix: Some post-secondary-in tending students will be expected to make connections among trigonometric functions, polar coordinates, complex numbers, and series



expected to

C29 analyse tables and graphs to distinguish between linear, quadratic, and exponential relationships (44, 46, 48,50,56,88, 100)

C30 describe and apply rates of change by analysing graphs, equations, and descriptions oflinear and

quadratic functions (72, 76, 78)

C31 analyse and describe the characteristics of quadratic functions (50, 54, 58)

C32 demonstrate an understanding of how parameter changes affect the graphs of quadratic functions (54, 58, 60)

C33 analyse and describe the characteristics of exponential and logarithmic functions (88, 100, 1 04)

C34 demonstrate an understanding of how parameter changes affect the graphs of exponential functions (90)

C35(Adv) write exponential functions in transformational form and as mapping rules to visualize and sketch graphs (102, 104)

C36 demonstrate an understanding of the relationship between angle rotation and the coordinates of a rotating point (138)

C37(Adv) describe and apply parameter changes within parametric equations of circles (140)

22 ATLANTIC CANADA MATHEMATICS CURRICULUM. MATHEMATICS 3204/3205

I

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INTRODUCTION

GCO 0: 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, using

appropriate units and using formulas (e.g., surface area, Pythagorean Theorem) and/or procedures (e.g.,

proportions) to determine measurements indirectly.

Key-Stage Curriculum Outcomes (KSCO) Specific Curriculum Outcomes (SCO)




expected to

KSCO i: measure quantities indirectly, using

techniques of algebra, geometry, and trigonometry

KSCO ii: determine measurements in a wide variety

of problem situations and determine specified degrees

of precision, accuracy, and error of measurements

KSCO iii: apply measurement formulas and

procedures in a wide variety of contexts

KSCO iv: demonstrate an understanding of the

meaning of area under a curve

Dl develop and apply formulas for distance and midpoint (126, 128, 134)


INTRODUCTION







angles formed by transversal intersecting parallel lines.

Key-Sta e Curriculum Outcomes (KSCO)



KSCO i: extend spatial sense in a variety of

mathematical contexts

KSCO ii: interpret and classifY geometric figures ,

translate between synthetic (Euclidean) and coordinate representations, and apply geometric

properties and relationships

KSCO iii: analyse and apply Euclidean transformations, including representing and applying

translations as vectors

KSCO iv: represent problem situations with

geometric models (including the use of trigonometric ratios and coordinate geometry) and apply properties

of figures

KSCO v: make and rest conjectures about, and deduce properties of and relationships between, two

and three-dimensional figures in multiple contexts

KSCO vi: demonstrate an understanding of the operation of axiomatic systems and the connections

among reasoning, justification, and proof

S ecific Curriculum Outcomes (SCO)


expected to

E3 write the equations of circles and ellipses in

transformational form and as mapping rules to

visualize and sketch graphs ( 136)

E4 apply properties of circles (116, 118, 122, 132,

134)

E5 apply inductive reasoning to make conjectures in geometricsituations (116, 118, 126)

E7 investigate and make and prove conjectures associated with chord properties of circles (116, 118,

120, 124, 136)

EB investigate and make and prove conjectures associated with angle relationships in circles (116,

120, 130)

E9 investigate and make and prove conjectures associated with tangent properties of circles (116,

130)

Ell write proofs using various axiomatic systems and

assess the validity of deductive arguments ( 116, 118, 122, 124, 126, 128, 130, 136)

El2 demonstrate an understanding of the concept of

converse (116, 118, 120)

24 ATLANTIC CANADA MATHEMATICS CURRICULUM. MATHEMATICS 3204/3205

INTRODUCTION







angles formed by transversal intersecting parallel lines.




KSCO vii: some post-secondary-intending students

will be expected to represent and apply vectors in

three dimensions, algebraically and geometrically

KSCO viii: some post-secondary-intending students

will be expected to explore and apply, using multiple

representations, circles, ellipses, and parabolas and, in

three-dimensional, spheres and ellipsoids



expected to

£13 analyse and translate between symbolic, graphical,

and written representations of circles and ellipses

(136)

£14 translate between different forms of equations of

circles and ellipses (136)

£15 solve problems involving the equations and

characteristics of circles and ellipses (132, 134, 136)

£16 demonstrate an understanding of the

transformational relationship between the circle and the ellipse (136)

ATLANTIC CANADA MATHEMATICS CURRICULUM: MATHEMATICS3204/3205 25

INTRODUCTION

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 and

issues, (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 role

with respect to statistical claims

KSCO ii: extend construction (both manually and via

appropriate technology) of a wide variety of data

displays

KSCO iii: use curve fitting to determine the

relationship between, and make predictions from , sets of data and be aware of bias in the interpretation

of results

KSCO iv: determine, interpret, and apply as

appropriate a wide variety of statistical measures and

distributions

KSCO v: design and conduct relevant statistical

experiments (e.g., projects with respect to current issues, career applications, and/or other disciplines)

and analyse and communicate the results using a

range of statistical arguments

KSCO vi: Some post-secondary-intending students will be expected to test hypotheses using appropriate

statistics



expected to

Fl analyse scatter plots, and determine, and apply the equations for curves of best fit, using appropriate technology (52, 98)

26 ATLANTIC CANADA MATHEMATICS CURRICULUM · MATHEMATICS 3204/3205

INTRODUCTION

GCO G: Students will represent and solve problems involving uncertainty.

Elaboration: Representing and solving problems involving uncertainty entails (i) determining probabilities by

conducting experiments and/or making theoretical calculations, (ii) designing simulations to determine probabilities in situations that do not lend themselves to direct experiment, and (iii) analysing problem

situations to decide how best to determine probabilities.




KSCO i: design and conduct experiments and/or

simulations to model and solve a wide variety of

relevant probability problems, and interpret and judge

the probabilistic arguments of others

KSCO ii: build and apply formal concepts and

techniques of theoretical probability (including the

use of permutations and combinations as counting

techniques)

KSCO iii: understand the differences among, and relative merits of, theoretical , experimental, and

simulation techniques

KSCO iv: relate probability and statistical situations

KSCO v: Some post-secondary-intending students will be expected to create and interpret discrete and continuous probability distributions and apply them in real-world situations



expected to

G1 develop and apply simulations to solve problems

(154, 164)

G2 demonstrate an understanding that determining

probability requires the quantifying of outcomes (144)

G3 demonstrate an understanding of the fundamental

counting principle and apply it to calculate probabilities of dependent and independent events (144, 146)

G4 apply area and tree diagrams to interpret and determine probabilities ( 148)

G5(Adv) determine conditional probabilities (150,

152)

G7 distinguish between situations that involve combinations and permutations (156, 160)

G8 develop and apply formulas to evaluate permutations and combinations (158, 160)

G9 demonstrate an understanding of binomial expansion and its connection to combinations (162)

G 10 connect Pascal's Triangle with combinatorial

coefficients (162)

G 11 (Adv) connect binomial expansions,

combinations, and the probability of binomial trials (164, 168)

G 12(Adv) demonstrate an understanding of and solve

problems using random variables and binomial distributions (166, 168)


INTRODUCTION

28 ATLANTIC CANADA MATHEMATICS CURRICULUM . MATHEMATICS 3204/3205

~ I

INTRODUCTION

Program Design and Components

Program Organization

The mathematics curriculum is designed to make a significant

contribution towards students' meeting each of the essential graduation

learnings (EGLs), with the communication and problem-solving EGLs

relating particularly well to the curriculum's unifYing ideas. (See the

Outcomes section of Foundation for the Atlantic Canada Mathematics Curriculum.) The foundation document identifies curriculum outcomes

at key stages of the student's school experience. Atlantic Canada Mathematics Curriculum Guide for Mathematics 320413205 presents

specific curriculum outcomes for the third of four courses in the grade

10-12 course sequence. These outcomes represent the means by which

students work toward accomplishing the key-stage curriculum outcomes,

the general curriculum outcomes, and ultimately, the essential graduation

learnings.

Outcomes Framework

Essential Graduation Learnings (EGLs)broad cross-curricular

i General Curriculum

Outcomes (GCOs)broad mathematical

expectations

Key-Stage Curriculum Outcomes (KSCOs)-at the end of grades 3, 6, 9,

and 12

i Specific Curriculum

Outcomes (SCOs)-for each grade level

Examples

Graduates will be able to use the listening, viewing, speaking, reading, and writing modes of

language(s) and mathematics and scientific concepts and symbols to think, learn, and

communicate effectively.

T contributes to

I

Graduates will explore, recognize, represent, and apply patterns and relationships, both informally

and formally.

T contributes to

I By the end of grade 12, students will be expected to represent functional relationships in multiple

ways (e.g., written descriptions, cables, equations, and graphs) and describe connections among

these representations.

T contributes to

I By the end of grade 12, students will be expected

to describe and translate between graphical, tabular, and written representations of quadratics

relationships


INTRODUCTION

Program Level

Advanced

Academic

Practical

It is important to emphasize that the presentation of the specific

curriculum outcomes in the guide follows a suggested teaching sequence.

Student and teacher resources have been developed to complement the

curriculum guide.

It is recognized that students' understandings of concepts will vary in

terms of depth and breadth. Curriculum and Evaluation Standards for

School Mathematics (NCTM, 1989) recommends that the study of

mathematics for every student revolve "around a core curriculum

differentiated by the depth and breadth of the treatment of the topics

and by the nature of the applications" (p. 9). While it is expected that all

students will work toward achievement of the same outcomes, it is

recognized that students will demonstrate different levels of performance.

Course 1 Course 2 Course 3 Course 4

Mathematics 2205 Mathematics 3205 Mathematics 3207 Mathematics 1204

Mathematics 2204 Mathematics 3204 Mathematics 3103

Mathematics 1206 Mathematics 2206 Mathematics 3206

Students wishing to take Level III will select either the Academic or the Advanced course depending on

their success in previous mathematics courses • their interests

30

their academic/career goals in and beyond high school

The high school mathematics curriculum has been moulded into four

courses, each designed for 110 h ours minim urn of instruction. All

students, at all program levels, will work toward achievement of the same

key-stage outcomes. While the key-stage curriculum ourcomes are

intended as targets for all students, all students will not be expected to

achieve them at a single level of performance. As well, there will be an

additional small percentage of students who will see their outcomes

significantly altered in individual educational programs.

Most of the specific curriculum ourcomes for each grade level are the

same, but not all. It is expected that some students can move from one

program level to another.

The students in the practical level courses will be expected to meet the

same key-stage curriculum outcomes and most of the same specific

(course) outcomes as those in academic and advanced. As well, the

instructional environment and philosophy should be the same at all

levels. The significant difference between practical and academic courses

is with respect to the level of performance expected in regard to each

outcome.

By and large, the practical course should be characterized by a greater

focus on concrete activities, models, and applications, with less emphasis

given to formalism, symbolism, computational or symbol-manipulating


Content Organization

INTRODUCTION

facility, and mathematical structure. Academic and advanced level courses

will involve greater attention to abstraction and more sophisticated

generalizations, while the practical course would see less rime spent on

complex exercises and connections with advanced mathematical ideas.

Typically, students who choose practical mathematics courses are those

who may have experienced considerable difficulty in mathematics

throughout their schooling 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 and may need

connections presented in more explicit ways. They often exhibit lower

self-esteem (in relation to mathematics) and require a pace that

accommodates the revisiting and reinforcing of concepts, skills, and

knowledge. These students need equal or perhaps greater access than

their peers to technology.

By way of a brief illustration, students at all levels should develop an

understanding of exponential relationships. Students raking x-level

courses have as much need as others to understand the nature of

exponential relationships, given the place of these relationships in

universal, everyday issues such as provincial and national debt and world

population dynamics. The nature of exponential relationships can be

developed through concrete, hands-on experiments and data analysis that

do not require a lot of formalism or symbol manipulation. The more

formal and symbolic operations on exponential relationships will be

much more prevalent in the academic and advanced courses. Students

who have career aspirations which involve the study of mathematics at

post-secondary should choose the advanced level of mathematics,

including Mathematics 3207. Other students who choose the academic level but still feel there is some chance that mathematics will be a

component of their post-secondary program should, at the very least,

include Mathematics 3103 in their high school program.

The N CTM Curriculum and Evaluation Standards for School Mathematics establishes mathematical problem solving, communication, reasoning,

and connections as central elements of the mathematics curriculum.

Foundation for the Atlantic Canada Mathematics Curriculum further

emphasizes these unifying ideas and presents them as being integral to all

aspects of the curriculum (see pages 7-11). Indeed, while the general

curriculum outcomes are organized around content strands, every

opportunity has been taken to infuse the key-stage curriculum outcomes

with one or more of the unifying ideas.

These unifying ideas serve to link the content to methodology. They

make it clear that mathematics is to be taught in a problem-solving

mode, that classroom activities and student assignments must be

structured to provide opportunities for students to communicate

mathematically, that teacher encouragement and questioning should


INTRODUCTION

Mathematical Modeling

Relations and Functions

32

enable students to explain and clarify their mathematical reasoning, and

that the mathematics with which students are involved on any given day

must be connected to other mathematics, other disciplines, and/or the

world around them.

The mathematical content identified in the strands must not be viewed

as independent units of study, but must be organized to develop depth as

well as breadth. For this depth to be developed, a number of common

connections must be visible to unify the core content. The unifying

connections are as follows.

Throughout the strands of mathematics that are being studied, students

need to see that mathematics is valuable in making predictions in the

real-world. Some basic mathematical structures that are used in modeling

include graphs, equations, tables, and algorithms. Students need to

understand also the limitations of modeling real situations, which are

most often very complex.

In some situations the modeling appears to be straightforward. Vectors

can be used to model an airplane in a wind current; exponential

functions can be used to model population growth; quadratics can be

used to model trajectory paths; and trigonometric functions can be used

to model wave motion. At other times the model may require

transformations in the data. Regression analysis will allow students to

better understand data from some real situations. In examining

population growth, logarithms plotted against time may produce a better

fit of the data to make predictions. Probability simulation may be used

to model processes involving gambling, insurance, and genetics.

An emphasis in the high school mathematics program is the study of

relationships between two quantities. Across all strands of the

mathematics program, students need to see the various ways in which

one quantity can vary in relation to another. This study will precede the

basic notion of function, how input and output are related, and how

functions may be described in various ways such as verbally, graphically,

algebraically, and numerically in tables. A formula such as the one for the

area of a circle, A(r) = 1t r-2, does not in itself provide a meaning of the

relationship. Students need to see how a change in the radius "r" results

in a corresponding change in the area 'A(r)'. This can be described

verbally or graphically.

The function concept is basic to the development of mathematical

thinking. The type of mathematical reasoning that has students

understand that the value of one variable may depend on the value of

another pervades all strands of mathematics. In trigonometry the

functions are periodic; in discrete mathematics the functions are not

continuous, but progress in steps; in exponential functions the functions


Communicating Mathematics

Multiple Representations

INTRODUCTI ON

are used to model growth and decay. In geometry students can examine

the function relationships that exist between the image and its object for

a given transformation, and in probability students may also view the

probability of an event as a function of the number of choices available.

Communicating in mathematics helps students to develop insight into

the nature of mathematics. Much of mathematics involves solving

problems where students are required to develop, interpret, and analyse

algorithms. When students are given a problem, they should be given

opportunities to share the various ways they solved the problem so that

they can compare the effectiveness, the efficiency, and the relative

appropriateness of the methods used. It is through this type of

communication that students deepen their understanding and extend

their ability to reason. Technology continues to advance, resulting in a

change in the type of problems that can be solved. It is important for

students to be able to communicate by using technology to solve

problems.

Mathematical arguments help students address questions such as How do

I know if I'm correct? Is this always true? Is there any solution to satisfY

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 and allows the

student to express ideas in many different ways, including discussions of

what is and is not accepted. Students may, for example, be asked to

clarifY when it is appropriate to use the exponential function to model a

situation.

A formal proof can illustrate the power of the axiomatic structure in

mathematics. Being able to move from examples to a deductive proof

convinces us of the truth of conjectures.

Mathematical discourse should be part of every lesson, since it promotes

both reasoning and understanding in mathematics.

Real understanding in mathematics is present when students are able to

use and choose representations to clarify and communicate. Students

who are in control of their learning may choose or find the

representation they find most useful. For example, a student who has

studied the quadratic function demonstrates mathematical power when

he or she is able to move between the graph and the equation to find

solutions to the quadratic equation or inequality and understands the

implications of these solutions.

An understanding of the multiple ways of representing an idea or solving

a problem, as well as recognition of the equivalence of the various

representations, results in deeper understandings of mathematical


INTRODUCTION

Learning and Teaching Mathematics

34

structure and process. For example, if students examine the same

geometry concepts from an Euclidean, analytic, and transformational

approach, they will develop a much stronger intuitive understanding of

these concepts.

What students learn is fundamentally connected to how they learn it.

The view oflearning mathematics as an integrated set of intellectual

tools for making sense of mathematical situations has created a need for

new forms of classroom organization, communication patterns, and

instructional strategies. The teacher is no longer the sole dispenser of

knowledge but is rather a facilitator and educational conductor whose

major roles include

• creating a classroom environment to support the teaching and learning of mathematics

• setting goals and selecting or creating mathematical tasks to help the students reach these goals

• stimulating and managing classroom discourse so that the students are clearer about what is being taught

• analysing student learning, the mathematical tasks , and the environment in order to make ongoing instructional decisions

Good mathematics teaching and learning take place in a range of

situations. Instructional settings and strategies should create a climate

that reflects the constructive, active view of the learning process. This

means that learning does not occur by passive absorption and imitation

bur rather as students actively assimilate new information and construct

their own meanings.

Students' opportunities to learn mathematics are a function of the

setting and the kinds of tasks and discourse in which they participate.

What students learn about particular concepts and procedures and their

own mathematical thinking depends upon the ways in which they

engage in mathematical activity in their classrooms. Their dispositions

toward mathematics are also shaped by such experiences. Consequently,

the goal of developing students' mathematical power requires careful

attention to pedagogy as well as to the curriculum.

Mathematics instruction should vary and should include opportunities

for group and individual assignments, discussion between teacher and

students and among students, appropriate project work, practice with

mathematical methods, and exposition by the teacher.

Instructional settings should include varied learning environments,

which encourage the development of specific co-operative behaviours.

Students should be expected to work together to help each other, and at

the same time they can be expected to complete individual projects.

Students develop strategies and skills in asking questions, listening,

showing and explaining to others how to do things, finding our what

others think, and determining a way to complete a project.


Summary of Changes in Instructional Practices

Integrating Technology

INTRODUCTION

Moving away from

•

•

•

teacher and text as exclusive sources of knowledge rote memorization of facts and procedures extended periods of individual practising of routine tasks instruction based almost completely on teacher exposition paper-and-pencil manipulative skill work the relegation of testing to an adjunct role with the sole purpose of assigning grades

toward practices that include

•

•

•

•

•

the active involvement of students in constructing and applying mathematical 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) the 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

The integration of computers, graphics calculators, video technology, and

other technologies into the mathematics classroom allows students to

•

•

•

•

explore individual or groups of related computations or functions quickly or easily create and explore numeric and geometric situations for the purpose of developing conjectures perform simulations of situations that would otherwise be impossible to 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 discovery learning and problem-solving

potential. At the same time, teachers have the opportunity to use

technology to communicate with fellow mathematics teachers, to share

lessons with experts, and to expose their students to information that

would otherwise be inaccessible.


INTRODUCTION

Learning Resources

I

36

L

Students will need to learn to make judgments as to when the use of

technology is appropriate and when it is not. It is very important that

mental math and computational estimation skills be fostered throughout

the senior high program. From time to time teachers should ask

students to perform basic computational tasks without the aid of

technology to ensure that basic skills are maintained. For example, on

working with exponents students should be able to determine 3

(% r Or solve Jogx 1000 = } for X mentally.

This curriculum guide represents the central resource for the teacher of

mathematics with respect to Mathematics 320413205 of the high school

mathematics program. Other resources are ancillary to it. This guide

should serve as the focal point for all daily, unit, and yearly planning, as

well as a reference point to determine the extent to which the curriculum

outcomes have been met.

Nevertheless, other resources will be significant in the mathematics

classroom. Textual and other print resources will be significant to the

extent to which they support the curriculum goals. Schools, school

districts, and Departments of Education should work together in

making professional resources available to teachers as they seek to broaden their instructional and mathematical skills. As well, manipulative

materials and appropriate access to technological resources need to be at hand.

It is highly recommended that teachers familiarize themselves not only

with Foundation for the Atlantic Canada Mathematics Curriculum, but

also with Curriculum and Evaluation Standards for School Mathematics, Pro fissional Standards for Teaching Mathematics, Assessment Standards for

School Mathematics, and Principals and Standards for School Mathematics (NCTM, April 2000). Because of the extent of information contained in

these documents, teachers are cautioned that assimilation of the ideas

contained will require much reflection, discussion, and rereading. All

high school mathematics teachers may wish to join the National Council

ofTeachers of Mathematics (NCTM) for professional growth.

Membership can include a subscription to The Mathematics Teacher, a

journal that contains a wealth of information and practical teaching

suggestions. Institutional or individual membership can be obtained by

telephoning 1-800-235-7566 (NCTM order office) or by contacting the

NCTM representative of your Mathematics/Science special interest council.


~

I

Assessing and Evaluating Student Learning

INTRODUCTION

In recent years there have been calls for change in the practices used to

assess and evaluate students' progress. Many factors have set the demands

for change in motion, including the following:

• new goals for mathematics education as outlined in Curriculum and Evaluation Standards for School Mathematics

The curriculum standards provide educators with specific information

about what students should be able to do in mathematics. These goals go far beyond learning a list of mathematical facts; instead, they

emphasize such competencies as creative and critical thinking, problem solving, working collaboratively, and managing one's own learning. Students are expected to be able to communicate mathematically, to solve and create problems, to use concepts to solve real-world applications, to integrate mathematics across disciplines, and to connect strands of mathematics. For the most part, assessments used in the past have not addressed these goals. New approaches to assessment are needed if we are to teach and address the goals set out in Curriculum and Evaluation Standards for School Mathematics.

• understanding the bonds linking teaching, learning, and assessment

Much of our understanding oflearning has been based on a theory that viewed learning as the accumulation of discrete skills. Cognitive views oflearning call for an active, constructive approach in which learners gain understanding by building their own knowledge and

developing connections between the facts and concepts. Problem solving and reasoning become the emphases rather than the acquisition of isolated facts. Conventional resting, which includes multiple choice or having students answer questions to determine if

they can recall the type of question and the procedure to be used, provides a window into one aspect of what a student has learned. Assessments that require students to solve problems, demonstrate skills, create products, and create portfolios of work reveal more about the student's understanding and reasoning of mathematics. If the goal is to have students develop reasoning and problem-solving competencies, 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 development of problem-solving and higher order thinking skills will become a realization only if assessment practices are in alignment with these goals.

In planning assessment, it is important to decide whether technology will be permitted. Certain assessment items become trivial when technology is used. It is recommended that since technology is an integral part of certain aspects of the curriculum, it should be permitted when those aspects are assessed. However, there may be rimes when assessment tools are created for use with others aspects of the curriculum where it is appropriate to decide that technology will not be permitted. It is important if students are permitted to use


INTRODUCTION

What Is Assessment?

Why Should We Assess Students?

38

technology during classroom activities that this be mirrored in assessment. Likewise, when the goal is for students to demonstrate mental facility, calculator use can interfere.

• limitations of the present methods used to determine student achievement

Does the present method of assessment provide the student with information on how to improve performance? We need to develop methods of assessment that provide us with accurate information about students' academic achievement and information to guide teachers in decision making to improve both learning and teaching.

Assessment allows teachers to communicate to students what activities

and learning outcomes they truly value. In order for teachers to assess

students effectively in a mathematics curriculum that emphasizes

applications and problem solving, teachers need to employ devices that

recognize the reasoning involved in the process as well as in the product.

AssessmentStandardsforSchoolMathematics (NCTM, 1995, p . 3) defines

assessment as "the process of gathering evidence about a student's

knowledge of, ability to use, and disposition toward, mathematics and of

making inferences from that evidence for a variety of purposes. "

Assessment can be informal or formal . Informal assessment occurs while

instruction is occurring. It is a mind-set, a daily activity that helps the

teacher answer the question, is what is taught being learned? Its primary

purpose is to collect information so that the teacher can make decisions

to improve instructional strategies. For m any teachers the strategy of

making annotated comments about a student's work is part of the

informal assessment. Assessment must do more than determine a score

for the student. It should do more than portray a level of performance. It should direct teachers' communication and actions. Assessment must

anticipate subsequent action.

Formal assessment requires the organization of an assessment event. In

the past, mathematics teachers may have restricted these events to

quizzes, tests, or exams. As the outcomes for mathematics education

broaden, it becomes more obvious that these assessment techniques

become more limited. Some educators would argue that informal

assessment provides better-quality information because it is in a context

that can be put to immediate use.

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

ATLANTIC CANADA MATHEMATICS CURRICULU M . MATHEMATICS 3204/3205

Assessment Strategies

Documenting classroom behaviours

Using a portfolio or student journal

INTRODUCTION

• inform parents of their child's progress so that they can provide more effective support

• certifY the level of achievement for each outcome

If we assume that assessment is integral to instruction and that it will

enable effective intervention in instruction, then it is essential that

teachers develop a repertoire of assessment strategies.

Some assessment strategies that teachers may employ include the

following.

In the past teachers have generally made observations of students'

persistence, systematic working, organization, accuracy, conjecturing,

modelling, creativity, and ability to communicate ideas, but often failed

to document them. Certainly the ability to manage the documentation

played a major part. Recording information signals to the student those

behaviours that are truly valued. Teachers should focus on recording only

significant events, which are those that represent a typical student's

behaviour or a situation where the student demonstrates new

understanding or a lack of understanding. Using a class list, teachers can

expect to record comments on approximately four students per class. The

use of an annotated class list allows the teacher to recognize where

students are having difficulties and identifY students who may be

spectators in the classroom. For summative purposes, grades should

reflect the degree to which students achieve the curriculum outcomes.

Having students assemble on a regular basis responses to various types of tasks is part of an effective assessment scheme. Responding to open

ended questions allows students to explore the bounds and the structure

of mathematical categories. As an example, students are given a triangle in which they know two sides, or an angle and a side and they are asked

to find out everything they know about the triangle. This is preferable to

asking students to find the side of a triangle in a trigonometry question,

because it is less prescriptive and allows students to explore the problem

in many different ways and gives them the opportunity to use many

different procedures and skills. Students should be monitoring their own

learning by being asked to reflect and write about questions such as

• What is the most interesting thing you learned in mathematics class this 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 of

the students' understand ing and progress as a problem solver. Students

should be doing problems that require varying lengths of time and


INTRODUCTION

Projects and investigative reports

Vvritten tests, quizzes, and exams

represent both individual and group effort. What is most important is

that teachers discuss with their peers what items are to be part of a

meaningful portfolio and that students also have some input into the

assembling of a portfolio.

Students will have opportunities to do projects at various times

throughout the year. For example, they may conduct a survey and do a

statistical report, they may do a project by reporting on the contribution

of a mathematician, or the project may involve building a three

dimensional shape. Students should also be given investigations in which

they learn new mathematical concepts on their own. Excellent materials

can be obtained from the National Council ofTeachers of Mathematics,

including the Student Math Notes. (These news bulletins can be

downloaded from the Internet.)

Written tests have been accused of being limited to assessing a student's

ability to recall and replicate mathematical facts and procedures. Some educators would argue that asking students to solve contrived

applications, usually within time limits, provides us with little knowledge of the students' understanding of mathematics. However, a test that is properly developed can be the most valid and reliable method

of collecting information about the degree to which students have

achieved the 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' understanding and comprehension as well as their procedural knowledge.

• Tests must be designed so that questions being asked reflect the expectations of the outcomes being addressed. Teachers must also reflect on the quality of the test being given to students. Are they being asked to evaluate, analyse, and synthesize information, or are they simply being asked to recall isolated facts from memory? Teachers should develop a table of specifications when designing their tests.

• In assessing students we have a professional obligation to ensure that the assessment reflects those skills and behaviours that we truly value. The bottom line is that good assessment is equivalent to good instruction and therefore promotes student achievement.

40 ATLANTIC CANADA MATHEMATICS CURRIC ULUM" MATHEMATICS 3204/3205

INTRODUCTION

Course Organization

Course Design

The Two-Page Spread

This section of the guide presents the Mathematics 320413205 mathematics curriculum outcomes that students are expected to achieve

during that year. Teachers are encouraged, however, to consider what

comes before and what follows to better understand how the students'

learnings at a particular course level are part of a bigger picture of

concept and skill development.

Mathematics 320413205 is organized into five units: Quadratics, Rate of

Change, Exponential Growth, Circle Geometry, and Probability. The

presentation of the specific curriculum outcomes in each unit reflects a

suggested teaching sequence. Certain outcomes have (Adv) written after

the outcome number. This identifies the outcome as one intended for

advanced level students, that is students of M athematics 3205. Many

other outcomes are addressed to a greater depth in Mathematics 3205

than in Mathematics 3204. This is addressed in column 2 and 3 of the

two-page spreads which follow.

The following pages derail curriculum outcomes. Each two-page spread

is dedicated to a small number of specific curriculum outcomes. As much

as possible, connections are made through references to other pages of

related outcomes or topics.

At the top of each page the overarching unit topic is presented, with the

appropriate specific curriculum outcome(s) (SCOs) displayed in the left

hand column. The second column presents the elaboration, which

includes instructional strategies and suggestions, as well as some examples

that might be used to illustrate achievement of outcomes. The third

column includes worthwhile tasks for instruction and/or assessment

purposes. While the strategies, suggestions, and examples are not

intended to be rigidly applied, they will help to further clarify the

specific curriculum outcome(s) and to illustrate ways to work toward the

outcome(s) while maintaining an emphasis on problem solving,

communication, reasoning, and connections.

The final column is entitled Suggested Resources and will , with your

additions, over time become a collection of useful references to resources

that are particularly valuable with respect to achieving the outcome(s)

given.

ATLANTIC CANADA MATHEMATI CS CUR RICULUM : MATHEMATI CS 3204/3205 41

INTRODUCTION

42 ATLANTIC CANADA MATHEMATICS CURRICULUM · MATHEMATICS 3204/3205

Unit 1 Quadratics

(20- 30 °/o)

QUADRATICS

Quadratics

Outcomes

SCO: In this course, students

will be expected to

C4 demonstrate an

understanding of patterns

that are arithmetic, power,

and geometric and relate

them to correspond ing

functions

C29 analyse tables and graphs

to d istinguish between linear,

quadratic, and exponential

relationships

C3 sketch graphs from

descriptions, tables, and

collected data

44

Elaboration -Instructional Strategies/Suggestions Students should extend previous knowledge to describe and reason in a variety of contexts using the mathematical constructs of function and relation and the symbolic language of algebra. Students should have had experience in creating and using symbolic and graphical representation of patterns, especially those tied to linear and quadratic growth. In this course these experiences will be extended to arithmetic, power, and geometric sequences with particular focus on quadratic and exponential relations.

C4/C29 To begin this unit, students should extend their work with patterns to include investigations of sequences of numbers that fall into two categories:

1) arithmetic sequences (a sequence where consecutive terms present a common difference)

2) power sequences, (a sequence made up of consecutive terms found by raising consecutive counting numbers to the same power) with a particular focus on quadratic relations

1 sr term (t I) IS 2+3 · = =3n-1

2"d term (t 2) IS 2+3· =5

3'd term (t 3) IS 2+3· =8

5'h term ( t 5) IS =14

100'h term ( t 100) IS 2+3· n'h term (t J IS 2+3· =3n - 1

Students should understand that an arithmetic sequence leads to a relationship that is linear and can be described symbolically. They might develop this in the following way: if given the sequence 2, 5, 8, 11, 14, find the lOO'h term. They can see that there is a common difference of three, and by completing an organized list (like that above) they might be able to predict the lOO'h term, then the nth term.

C4/C29/C 3 Students should examine sequences through interplay between various representations (verbal, symbolic, contextual, pictorial and concrete). For example, students might be given the diagram of towers (which represents the sequence above) and be asked to construct the towers with cubes and record the number of cubes in each tower as a sequence. They could graph the number of cubes in each tower against the tower number and examine this relationship. Students could talk about the constant growth rate, since the height increases by three with each additional tower. They should relate the constant growth rate to the slope of the line that would pass through the plotted points. Students might be asked to describe, in words and symbolically, the relationship between the tower number in the sequence and the corresponding height. From their description they might predict the 1 O'h term in the sequence.

As a way of checking their equation, students could use graphing technology to obtain the equation. To do this, for example, they could enter the sequence of tower numbers in one list, and the sequence that represents the height of each tower into a second list, and use linear regression capabilities to obtain the equation y = 3x- 1.

. .. continued


~

I

Quadratics

Worthwhile Tasks for Instruction and/or Assessment

Pencil and Paper (C4/C29/C3)

1) Complete each sequence and find the n'h term. a) 2, 4, 6, 8, 10, _ , _ , _ , ... n 'h .

b) 3, 6, 9, 12, _, _, ... n 'h.

c) 17, 12, 7, 2, ... n'h.

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

Performance (C4/C29/C3)

3) Explore the following dot patterns and determine if they form an arithmetic sequence: If the pattern is arithmetic, complete the table given to find the 1 O'h and 20'h term.

a)

. • •

. . . . . . . . . . . b)

. . ••

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

N umber of dots in array: s Number of dots in array: t

4) If this graph represents a sequence of numbers:

a) Is the sequence arithmetic? b) What would be the value of the 8'h term? c) Describe the sequence in words d) Describe the n'h term e) Write the equation for the graph.

Q; 16 ..0

§ 12 c Cl

·E s c ::::l

8 4

2 3 4 5

term number 5) Create a problem situation that can be represented with an arithmetic sequence.

Have the problem solver display the sequence graphically, concretely, and symbolically for a purpose.

ATLANTIC CANADA MATH EMATICS CURRICULUM MATHEMATICS 3204/3205

QUADRATICS

Suggested Resources

45

QUADRATICS

Quadratics

Outcomes


will be expected to

C4 demonstrate an



and geometric, and relate

them to corresponding

functions


to distinguish between linear,


relationships

46

Elaboration -Instructional Strategies/Suggestions

C4/C29 As students examine sequences they will sometimes notice that the difference

between the terms in the sequence does not increase/decrease at the same rate. For

example, when students

examine the dot pattern given

below and create the sequence

1, 4, 9, 25, 36, ... to represent

the number of dots in each

figure, they should notice that

2 3 4

each term in the sequence is the square of the number of dots on one side. They could

use this to determine an algebraic expression (tn = n2) from which they could predict

other terms in the sequence.

They should be able to describe the difference between the above pattern and the

arithmetic patterns looked at previously. In the sequence above, there is no common

difference between successive terms. The differences are 3, 5, and 7. However, if the

consecutive differences are subtracted, students would see

a common difference occur at the second level. When a to-+ 1 4 9 16 25 36

common difference occurs at the second level, a second

degree (quadratic) equation will result. The equation

describes the quadratic relationship between the term and

v v v D, -+ 3 5 7

v v v D, -+ 2 2 2

v 9

v 2

v 11

the term number. Sequences with a common difference that does not occur at the

first level are called power sequences. The power sequence described above is

quadratic.

Students might use cubes to build towers and compare the growth rate visually

between a quadratic sequence and that of an arithmetic sequence.

Teachers could ask students to extend the pattern below by drawing a sketch of the

next figure, then have them build a table like the one below.

#cubes Total along #of

one edge cubes D, D, Dl

® > 7

2 8 > 12 > 19 > 6

GJ 3 27 > 18 > 37 > 6

4 64 > 24 > 67

5 125

When a common difference occurs at the third level, a third degree (cubic) equation

will result. This is another example of a power seq uence.

ATLANTIC CANADA MATHEMATICS CURRICU LUM MATHEMATICS 3204/3205

Quadratics


Performance (C4/C29)

1) The area of a shape is the number of units inside. The figures below illustrate the areas of a shape that continues to grow.

§ ~d··· 1 2 3 4 5 6

a) Copy and complete the following number sequence for the areas of the shapes above.

3, 6, _,_, _,_, ... b) What kind of sequence do the areas form? Describe the relationship between

the number of square tiles along the bottom edge and the numbers in the sequence.

c) Copy and complete the number sequence of the perimeters

8, 12, -'-' _, ... d) What kind of sequence do the perimeters form? Explain . e) Predict the area when there are 25 units along the bottom edge. What would be

the perimeter for this same shape?

2) These figures illustrate a sequence of squares in which the length of the side is

successively doubled.

a) What are the perimeters of these four squares? b) What happens to rhe perimeter of a square if the length of irs 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 irs side is doubled?

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

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

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

journal (C 4/C2 9)

4) Teachers could ask students to explain how finding common differences in patterns helps determine what power equation to use.

ATLANTIC CANADA MATHEMATICS CURRICULUM : MATHEMATICS 3204/3205

QUADRATICS

Suggested Resources

47

QUADRATICS

Quadratics

Outcomes SCO: In this course, students

will be expected to

CI model real-world

phenomena using quadratic

functions

C8 describe and translate

between graphical , tabular,

written, and symbolic

representations of quadratic

relationships




relationships

48

Elaboration -Instructional Strategies/Suggestions Cl Throughout this course, students will use mathematics to solve practical

problems. Often, students will need to simplify the problem, determine irrelevant

information, and express the problem with mathematical language or symbols. This

is called making a mathematical model.

For the problem in column three about making boxes to hold popcorn, a variery of

different mathematical models is possible. For example, students might draw a scale

diagram, create a table, or draw a graph to show the relationship. Finally, they may

describe the relationship algebraically. Each of these is a mathematical model of the

problem.

In previous studies students have analysed and applied linear functions and have

come to understand that a linear relation represents a constant growth rate. For

example, in a snow board rental scheme, for every hour of rental, the cost increases by

$3.50. When students graph this relationship, they can see that as the points progress

from left to right on the graph, the cost increases by $3.50, and because this growth is constant, it represents a discrete linear relationship.

Students might remember from previous activities that, with a fixed perimeter, as the

dimensions of a rectangle approach a square the area increases, maximizing with the

area of the square. In the campsite problem below, however, they are working with

string for three side lengths of a rectangluar campsite, the fourth side being a river

bank.

The campsite problem asks campers to stake out their campsite with 50 metres of

string. They are to create a rectangular boundary, but since one side is along a river

bank no string for that side is necessary. Students

must find the length and width measurements to

maximize the rectangular area of their campsite.

C8/C29 For the campsite problem, students

might be asked to create a table of width versus

area, beginning with a width of 5 m . As they

record the different area calculations, they should

notice that even though the width is increasing at a

constant rate, the rate of increase is slowing down

width

5 6 7 8 9 10 11 12 13

length area 0, o,

200 > 28 228 > 24 > -4 252 > 20 > -4 272 > 16 > -4 288 > 12 > -4 300 > 8 > -4 308 > 4 > -4 312 > 0 > -4 312

until it reaches 312 m 2• These decreasing differences are evident in the first set of

differences (D1), and this set of differences changes at a constant rate (evident in the

second set of differences (D2

) ) . This constant in D2

occurs when the relationship is

quadratic.

Spreadsheets and/or table features on calculators might be used to generate the values

in the table.

Formulas like "50- 2x" for length and "x (50- 2x)" for area would be necessary in

this example.

Students should be encouraged to describe how the width/area pattern in this table

differs from a table that shows a linear relationship.


Quadratics

Worthwhile Tasks for Instruction and/or Assessment Pencil and Paper (C29)

1) Compare the data in these tables. Which table(s) do you think represent(s) a quadratic relationship and explain your thinking.

n l1234 567

f(n) 90 75 60 45 30 15 0

10 20 30 40 50

f(x) 9 11 16 20 30 25

( I o 2 3 4 5 6 7 8 9 10

f(r) 0 10 18 24 28 30 30 28 24 18 10

Performance (Cl/C8/C29)

2) Murray and his friends were sitting in a booth at The 01' Pizza Parlour that specializes in pizza pies a Ia 1960. Their menu advertises the following prices for plain cheese pizza:

Small: (8inch diameter) $0.85 (lOinch diameter) $1.15

Medium: (12inch diameter) $1.55 (13inch diameter) $1.75

Large: (lSinch diameter) $2.25 (18inch diameter) $3.15 (24inch diameter) $5.50

a) Murray wonders if there is a mathematical relationship between the diameter and the price. Create a table and explain to Murray what the relationship is.

b) Create a graph for the relationship. c) IfThe 01' Pizza Parlour offered 20 inch pizzas, what do you predict would be the

price? d) If the parlour wanted to introduce a gigantic size and sell it for $7 .50 , what would be

the diameter of the gigantic? e) Based on your model , what does the price-intercept represent?

Performance (Cl!C8/C29)

3) Swimming Pool Problem: In the design of a particular swimming pool and surrounding patio (which we will call "square pools") the water surface of each pool is in the shape of a square. Around the pool there is a border of square-shaped patio tiles. The pools come in many sizes. Here are pictures of the three smallest pools available.

a) Using the two-sided square tiles (such as unit tiles for algebra), build models of the above pools, using red for the water surface and white for the patio tiles. i) Organize data into this table.

ii) Plot graphs of Pool # # of red tiles # of white tiles total # of tiles

A) pool # vs. # of red tiles B) pool # vs. # of white tiles

iii) Describe the shape of the two graphs. How are they the same? How are they different?

A) Using cubes build towers to model how the growth in columns 2 and 3 changes.

B) When will the number of red riles and the number of white riles be the same? Explain


QUADRATICS

Suggested Resources

Barnes, Mary, Investigating

Change: Functions and

Modelling. Unit I.

Melbourne: Key Curriculum

Corporation, 1992.

Swan, Malcom. The Language

of Functions and Graphs.

Nottingham, UK: Shell

Centre for Mathematical

Education, 1985.

49

OUADRJI.TICS

Quadratics


will be expected to

Cl model real-world


functions



collected data


between graphical, tabular,



relationships




relationships

C31 analyse and describe the

characteristics of quadratic

functions

A7 describe and interpret

domains and ranges using set

notation

50


Cl/C3/C8/C29 Continuing with the campsite problem from the last page, students

should produce a graph from the table of data they have calculated. As they plot the

data points, they should notice that the points do 316 not fall along a straight line. They should discuss 308

how this shows that the area does not increase at a

constant rate as it approaches the vertex. They will

see as they continue to plot points that the graph

takes on a nonlinear, parabolic shape. They will

need to extend their table from the previous page

to include larger values for the width that will result

ro 300 Q)

«; 292

284

. .

910111213141516

width (m)

in the area decreasing. Students should discuss why the area begins to decrease.

In analysing the graph, students should discuss whether or not it is correct to join the

data points and, if they did, what these additional points would represent.

C31 Students should notice the symmetrical shape of the graph and understand that chis symmetry is expected in a quadratic function. They should also note how the

symmetry is visible in the table. They should explore, using the trace feature, the

maximum value for the function with respect to the problem.

C311A7 For the graph above, if a line were drawn through the maximum point (the

vertex) parallel to the vertical axis, this line would be the axis of symmetry for the

graph of the quadratic relationship. The equation of the axis of symmetry is

symbolized as x = "x-coordinate of vertex." They-coordinate of the vertex is called the

maximum value for the function and represents the maximum area for the campsite.

The x-coordinate of the vertex represents the width that results in the maximum area.

When asked for an appropriate domain for the graph in this situation, students might

respond by saying "x-values from 1 to 20." They should symbolize this as

{1 ~ x ~ 20, xE R}. Students may use interval notation, x E [1,20], to represent the

same thing. Students should discuss how the maximum (in this case) and minimum

values determine restrictions on the range and how these are written symbolically. For

example, in this problem the maximum value is 312.5 m and there is no value below

zero. The range is {0 < y::; 312.51 y E R} or yE (0,312.5) ·

Students should be reminded that the model quadratic function is

y = :x? whose domain is xE R, and whose range is{y 2: oJ yE R }. These can both be

written in interval rotation: domain: x E R ( -oo , oo ), range: y E [0, oo) The graph of this

quadratic function is symmetric about a line drawn through the vertex parallel to the

vertical axis, and this symmetry can be

seen in a table of values.

Students should be reminded that the

following pattern occurs in the graph of y = :x?:

right/left

1 2 3

up

1 4 9

-1

From the vertex, over I, up 1; from the vertex, over 2, up 4; .from the vertex, over 3, up 9 .

This pattern is useful for sketching quadratic graphs and their transformations.


Quadratics


Journal (C8/C29/C31)

1) Explain how you can tell by examining this table of values that the equation that contains these points will be quadratic. Describe the graph of this data.

X ~-2 -1 0 1 2 3 4 y -1 -10-15-16-13 -6 5

Peiformance (CI/C3/C8/C29/C31/A7)

2) Begin with two cubes snapped together to form a rectangular prism ([I]). Add cubes to model the following pattern.

n roral number of cubes I I H I I

ODD II an I DO DOD OCTTJ first second third

1 2 3 4 5

10

25

2 6

a) Count the total number of cubes used for each pattern. Complete the chart. Describe an appropriate domain and range for this relationship.

b) What kind of a relationship comes from the pattern? c) Graph the relationship. Should you join the

points? Explain. d) Is there a maximum or minimum point? What

is the axis of symmetry equation? Explain.

3) How does the speed of the ball change as it flies through the air in this golf shot? a) Discuss this situation with your neighbour, and

write down a clear description stating how you both think the speed of the golf ball changes.

b) Sketch a rough graph to illustrate the relationship between the height of the ball and time in seconds. Compare your graph with the given drawing. How is it the same? How is it different? State and interpret an appropriate domain and range for this relationship.

(ij .0 Q)

£ 0 "C Q) Q) Q.

"' time after the ball is hit


QUADRATICS

Suggested Resources

51

QUADRATICS

Quadratics

Outcomes


will be expected to

Cl model real-world


functions

Fl analyse, determine, and

apply scatter plots and

determine the equation for

curves of best fit , using

appropriate technology



collected data

C23 solve problems involving

quadratic equations

52


Cl Students should, through a variety of experiences with functions, come to recognize the elements in a real-world problem that suggest a particular model. For example, area, accelerated motion, and trajectory suggest quadratic functions. There are two types of situations that students will encounter. One is the issue of finding the maximum or minimum value that does not necessarily require the solving of an equation (discussed on this page). The other involves finding the intercepts, or zeros, as in the type of problem where students need to determine a value for a dimension that results in the doubling of an area. This does require the solving of an equation (discussion beginning on p. 56) .

Fl/C3 In their previous study students will have conducted experiments and explored data that was best fit by linear models. Sometimes the data was nonlinear and curved. Students explored it with power, exponential, and quadratic regression using technology.

In this course, students might conduct experiments and gather and plot data that will result in quadratic equations. For example, they might roll a pulley along a wire with a downward slope that has been marked in 1 0-cm intervals. Students would time the pulley from the starting position to each marked interval. They would use this data to determine an equation.

Teachers may also want to use a calculator based laboratory unit that connects to a graphing calculator or a computer-based laboratory to conduct experiments and collect data. Once the data is collected and displayed, it can be analysed to produce a mathematical model. For example, a motion detector can be used to gather data on a bouncing ball. The graph produced might resemble the one shown. Each of the bounces is a parabola. Students can use the technology to find the equation that best represents the data for each separate parabolic part of the graph and can compare the coefficients in each of the equations.

C23 There are geometric structures where the relationship between the dimensions and the area is quadratic. For example, students might explore irregular rectangular shapes, as in the diagram, and find the values for x andy that would maximize the area. They should determine that the area is A = 4.>? + (-9.>? + 3x + 2).

X

n-(2~xl_ , 4x~

When solving problems involving quadratic equations, students should observe that different strategies are required for solving the problem depending on the different situations described in the context. Students should solve problems that can be modeled with quadratic equations. For example, an addition of uniform width is being attached to both the front and right side of a one-storey building. If the building measures 15.5 m by 11 .3 m, determine the width of the addition that results in doubling the floor area. Students should see that the strategy of solving an equation is required.

155m

11 3m

X I

I

In the campsite problem, seen earlier, a maximum value is required. This can be determined without solving an equation. Another problem might involve trajectory paths. Objects shot or thrown into the air often follow a path that can be described with a quadratic equation. For example, an arrow is shot into the sky along the trajectory 16? + 60t + 1.5 = h, where "h" is the height in metres and "t" is the time in seconds. Students could be asked to find the maximum height. Manipulating the equation into transformational form could be helpful. Students could be asked to determine how long the arrow was in the air. This might require them to solve an equation. Students might also be asked to evaluate (rather than solve) the trajectory equation for various heights given time, or for various times, given heights.


Quadratics

Worthwhile Tasks for Instruction and/or Assessment Performance (C l!Fl!C3/C23)

1) Tell students to practise rolling a large marble or ball bearing on a propped-up clipboard from the lower left so that it rolls up the board and back down, leaving at the lower right. When they are ready, tell them to do the following]: a) Tape a sheet of carbon paper over a sheet of graph paper so that the marble will

trace a path onto the graph paper. b) Roll the marble one more time, remove the carbon paper, and draw the X- and

Y- axis on the graph paper in an appropriate place. c) Carefully locate several points on the curve and find the equation that best fits

the data. d) Interpret the data. e) Determine the equation. f) Analyse the graph or use the equation to predict how far up the board the

marble was after 0.05 seconds, 0.62 seconds. g) Explain how long the marble stayed on the board. h) Find two times when the marble was the same height up the board.

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

timeinseconds 15 25 48 60 71 100 120 130 150 180 190

water (L) remaining 55 51.1 42.6 38.6 3 5 26.5 21.4 18.8 14.6 9.5 7.9

Have students a) sketch a graph of the situation b) from the graph, predict how long it takes to empty the tub c) determine what function might model this situation and explain

Journal (C3/ Fl)

3) When students conducted the Ball Bounce activity in the classroom they observed (with a motion detector) a ball bouncing straight up and down beneath the detector. Yet the graph produced seemed to depict a ball bouncing sideways. Explain why this is so.

4) From the above activity, the graph produced appeared to be made up of many adjoined parabolic shapes. Focus on any one of those parabolic shapes and explain why it can be modeled by an equation in this form y = a (x- h) 2 + k. Explain how you would use this to determine the actual equation.


QUADRATICS

Suggested Resources Brueningsen, Chris et al.

Real-World Math with the

CBL System. Dallas: Texas

Instruments, 1994.

53

QUADRATICS

Quadratics

Outcomes


will be expected to

Cl model real-world


functions





relationships



functions

C32 demonstrate an

understanding of how the

parameter changes affect

graphs of quadratic functions

54

Elaboration -Instructional Strategies/Suggestions Cl/C8/C31/C32 In previous studies students examined the transformations of a quadratic function and how those transformations are visible in the equation. Thus, students should be

able to determine the equation of a quadratic using reverse logic. For example:

When students see a quadratic relation, as in the graph, the maximum point tells them the vertex coordinates, which gives them the horizontal and vertical translation values. Thus the equation of y = >!- , becomes

y -5=(x-2Y .

Students must then decide if there is a vertical stretch, and if the parabolic shape is a reflection of y =>!-in the x-axis. Because there is a maximum value, there is a reflection in the x- axis, and the equation becomes - (y- 5) = ( x- 2 Y . It seems most natural to deal with the translation first, then the reflection, and then investigate whether there is a stretch or not. The pattern between the points on the graph should be from the vertex over 1 down 1 (which appears to be true); from the vertex over 2 down 4 (the square of2). This is true (based on they-intercept), so the quadratic relation is not stretched, and the final equation forthisgraphis -(y-5)=(x-2f.

Sometimes the stretch factor is hard to predict. Students can calculate the stretch factor if they know the vertex and one other ordered pair. For example, a baseball is thrown into the air to pass through a loop 20m above the ground. If it passes through the loop at the top of

its path and is caught 10m from a point directly below the loop, determine the equation that describes its path. Since the students know about symmetry, the point directly below the loop must be halfway between where the ball was thrown and where it was caught. The vertex point has a value (1 0 , 20), assuming (0, 0) to be the point where the ball is thrown. The equation is k (y- 20) = ( x -l 0 )

2 •

Another point on the curve would be (20, 0) . Thus k(o- 20) = (20-10r.

-20k = 102

k= 100 =-5 -20

Students can now solve fork .

1 Students should conclude that since the value fork is -5, the stretch is 5, and there is a

reflection in the x- axis. Students will write the equation of the path as -5 (y- 20) = (x- 10)2

•

When the vertex point is not known, but three other points appear to be part of a parabolic curve, students can use a system of equations with three variables (learned in a previous course) to obtain the quadratic equation.

Since students know the function to be modelled is quadratic, they would use the general form for a quadratic (y =CD?+ bx +c) to model their system. Given the three points, they can determine three equations with variables a, b, and c. When they solve the system, they will have values for a, b, and c and can substitute these values into the general form to state the equation.

Note: Some students may have seen this procedure when they studied the solving of"3 by 3" systems of equations in a previous course. Other students may be studying this course first and will learn that procedure later when studying the other course.


Quadratics


Pencil and Paper (Cl/C8/C32)

I) An experiment was conducted and the following data was collected and put in a table.

Time (seconds) 2 .0 2.5 3.0 3.5 4.0

Height (metres) 8.0 28.0 32.0 35.0 32.0

Your graphing calculator is broken. Use the table to determine the equation that best represents this data.

2) State the equations of the following graphs.

i)

iii)

Perfonnance (C8/C31/C32)

2

ii)

iv) 4

3

3) How does the following equation allow you to visualize its graphical representation?

_ _!_(y -1) = (x + 5) 2

2

4) A rocket attains a height of 250 m when it is fired at a target that is 200 m distant on the horizon. There is a building, 85 m high, 20m from the target and on direct line of fire. Will the rocket reach the target, and if so, by how m uch will it clear the building?


QUADRATICS

Suggested Resources

McKill ip, David. Pre-Calculus

Mathematics One. Toronto:

Nelson, 1992.

55

QUADRATICS

Quadratics

Outcomes


will be expected to

Cl O(Adv) determine the

equation of a quadratic

function using finite

differences




relationships

56

Elaboration -Instructional Strategies/Suggestions C 1 O(Adv) I C29 Sometimes the coordinates of the vertex cannot be read directly from a

given set of data. For example, from the given table students would know that the data is not linear (no common

x123456 y 5 19 43 77 121 175 difference (D

1) between they-values and the x- values

vvvvv )H h · o, 14 24 34 44 54 increase at a constant rate . owever, t ere rs a common

02 v v v v 10 10 10 10

difference at the second level (D2 )This implies that this relationship can be modelled using a second degree equation (quadratic). Students can tell from the table that the D

1

values are decreasing in moving to the left. To approximate the vertex, students will need to examine where D

1 approaches zero.

Students in the enhanced course should be able to find the equation of the quadratic relation using the finite difference method. For example, when students compare the table above with the table for the general quadratic equation y = ax2 + bx + c below, they can create equations with variables a, b, and c. a ::F 0

X 2 3 y a + b + c 4a + 2b + c 9a + 3b + c

D, ~ ~

D, ~

2a

Note: the x-values must have an increment of one. Students should use the fact that the 2a is in the place of the 10 from the first table. Students would then solve 2a = 10, to get a value for "a": a= 5.

Using the first level differences,(D 1

) •

The 3a + b aligns with the 14 -7 3a + b = 14

15+b=I4

b = -1

The a+ b + c aligns with the 5 - a+ b + c = 5 5-1+c=5

c= 1

Substituting the values a= 5, b = -1, and c = 1 into the model y =ax?+ bx + c gives the equation y = Sxl- x + 1. For various reasons, students might choose an alternative method to get the equation. They have observed that the constant in the second level of differences is twice the value of the coefficient a in the related quadratic (D 2 )

equation. Substituting the calculated value for a into the general form of the equation and using two of the known coordinate pairs, students can establish a "2 by 2" system of equations and solve for the band c values. Values for a, b, and c can then be substituted into the general form to obtain the quadratic equation needed. This method is most often used because the

Given in the above that

2a = 10, then a= 5, so

substituting:

5 = 5(1 )2 + b(l ) + c

19 = 5 (2) 2 + b(2) + c

different table for the general quadratic (middle of page) is hard to remember and, more importantly, can be used only when independent variable values increase by one.


Quadratics


Performance (CIO(Adv)/C29)

1) The picture to the right simulates an object dropping from a height of 10 m. The height is recorded every 0.20 sec. Carefully determine, and record in a table, the heights of the ball at each interval, accurate to the nearest 0.01 m. a) Find the equation that describes the height of the object versus

nme. b) After how much time will the object hit the ground? c) How high will the object be 0.5 seconds after being dropped?

2) a) Find an equation that expresses the relationship between the number of sides of a polygon and the number of diagonals formed by connecting the non-consecutive vertices.

b) Find the number of diagonals for a polygon with 20 sides.

3) A corner store charges $2 for a pack of AAA batteries. On average, 200 packs are sold each day. A survey indicates that the sales will decrease by an average of 5 packs per day for each 1 0-cent increase in price. a) Use the information given to complete the row labelled

"number sold."

Selling Price 2.00 2 .10 2 .20 2.30 2.40

Number sold 200

Revenue

Dl I I I I I D

b) Calculate the revenue using the selling price and the number sold. Then calculate the first and second differences.

10

9

8

7

6

5

4

3

2

0 0 2

c) Write an equation that describes the relationship between the revenue, y, and the selling price, x , charged per pack.

d) Graph the equation and find the maximum revenue. What selling price provides maximum revenue?


QUADRATICS

Suggested Resources

57

QUADRATICS

Quadratics


will be expected to

C9 translate between different

forms of quadratic equations



functions



notation

C32 demonstrate an

understanding of how the

parameter changes affect the


58

Elaboration -Instructional Strategies/Suggestions C9 Quadratic equations are written in a variety of forms. These include the general form

y = ax2 + bx + c, a * 0 ; a transformational form ( 1- (y - v) = ( x - h )2

) ; and standard form

y = k(x- h) 2 + v) . In context, quadratic equations are usually written in general form. For example, students may be told that a football, when kicked, follows the path h = -?- + 15t + 3, or in another context, they may obtain an equation using quadratic regression that looks like y = 20.1x2- 331.5 x + 121.5. However, obtaining equations from graphs or tables, students will

often state the equation in transformational form -~ (y -1) = ( x + 5 f or in standard form y

= -2(x + 5) 2 + 1. Students should be able to move from one form to the other and appreciate why they need the equation in a particular form. For example, students may want the equation in transformational form to get the maximum value.

Assuming thatj(x) = -2x2 + 16xdescribes the path of a projectile, where "x" is the time in seconds and j(x) is the height in metres, students are asked to find its maximum height.

To put an equation into transformational form or standard form requires an algebraic procedure called "completing the square." See C9/B 1 elaboration on the next two-page spread.

C31/A7 /C32 When the equation is in transformational form or standard form, students can not only see the maximum or minimum value, but can give a complete analysis of the function since each of the transformations visible in the equation affects the shape and position of the parabola. The maximum/minimum value comes from the vertical translation value, which, in turn, comes from the vertex. The vertex helps inform students about an appropriate window when using graphing technology. The vertex also defines the range (since the axis of symmetry

passes through the vertex). For example, -~(y -1 )= (x + 5f has as its graph:

analysis

vertex: (-5, 1)

axis of symmetry: x =

domain: {xE R}

range: {y ~ 1lyE R}

maximum value: 1

y-intercepr: _...!_ (y- 1) = 52

2 y-1= -50

y=-49

-1 -1

zeros: 1 2 --(-1)=(x+5)

2 1 2 -=(x+5) 2

J2 x = -5 ±- (the exact roots)

2 :. x B -5.7, and -4.3 (the approximate roots)

Students will often want the equation in standard form, especially when using technology. They could easily express as y = -2(x + 5)2 + 1.

ATLANTIC CANADA MATHEMATICS CURRICULUM. MATHEMATICS 3204/3205

Quadratics

Worthwhile Tasks for Instruction and/or Assessment Peiformance(C9/C31/A7/C32)

1) Remind students about the various properties (vertex, axis of symmetry, domain, and range and maximum value, the y-intercept, and number and values of zeros) of a quadratic function and that these properties are visible in an equation. Ask them to

begin with j(x) = -2x2 + 12x + 3 and determine the various properties. 2) Given the following properties, have students determine the equation and draw the

graph of the function.

Properties:

vertex: (-0.8333, 14.08333) axis of symmetry: x = -0.8333 domain: {xE R} range: {y E Rl y :S 14.08333} y-intercept: 12 zeros: {-3, 1.333}

3) Create questions like 1) and 2) above and the corresponding answers. Interchange with your group members and have them answer the questions.

4) Given the following equations in transformational form :

a) 3 (J - 2) = (X + 1 y b) ~ (J + 3) = (X - 5 y 2

c) -5(y-8)=(x +3/ d) -}y =(x-1)

i) Describe the transformations of y = x2 for each that are visible in the equation.

ii) Sketch a graph on grid-paper that approximates the location and shape of each.

iii) Change each equation into standard form. iv) Describe how the transformations of y = x2 are visible in the standard form

of the equation . v) Enter each equation into your graphing calculator and graph to check your

approximations in part b).

Journal (C9/C311A7/C32)

5) Describe how the properties of a quadratic relationship can help you to determine the equation for that relationship. What are the minimum number of properties required to determine the equation. Which properties are they?

Journal (C32)

6) Explain why the graph of y = 3(x-2)2 looks like there could be a horizontal stretch even though it is not visible in y = 3(x- 2)2

• Show how you could write a quadratic equation in transformational or standard form to show the horizontal stretch.

1 7) A quadratic equation has a horizontal stretch of l, and no vertical stretch. Ben does

not like horizontal stretches. Show how he could write the same equation to show the corresponding vertical stretch so that the graph would look the same as that drawn by a

1 student using a horizontal stretch of l .

ATLANTIC CANADA MATHEMATICS CURRICULUM · MATHEMATICS 3204/3205

QUADRATICS

Suggested Resources

59

QUADRATICS

Quadratics

Outcomes


will be expected to

C9 translate between different

forms of quadratic equations


quadratic equations

Bl demonstrate an

understanding of the

relationships that exist

between the arithmetic

operations and the operations

used when solving equations

C32 demonstrate an

understanding of how



60

Elaboration -Instructional Strategies/Suggestions C9/C23/Bl For graph sketching, or for finding maximum/minimum values in a problem, students should translate a quadratic equation into transformational form or standard form . In these forms the right-hand side of the equation is expressed as a factored perfect-square trinomial. Students should use concrete materials to develop an understanding of completing the square and to help make sense of the symbol manipulation.

They would begin with the given form : , • .............. 3~ ..•. .\o.. ......... i

Using algebra tiles, students would construct a rectangle from the expression -2x2+ 16x.

!_,.,.,.,

······· ·······-------------------------·

To create a square: first, students should use the strategy "make it simpler," so that the coefficient of x

= 1, and remove the upper half of the rectangle (divide by two).

~ Y = Dllllllll -x2

+ax

Secondly, they should change the white x? tile to a positive value and the x tiles to negative values (multiply all terms by-1).

Thirdly, to facilitate completing the square, students should move four -x tiles to a position above the square tile

and "complete the square" by filling in the space with the appropriate number of units. Students should notice the 16 is added to both sides.

x2 -8x+ 16

It is important for students to understand that the " 16" is obtained by taking half the number of x's to form the square, requiring a "4 by 4 " set of units to complete the square.

1 2 To complete the symbol work, students should record the perfect -- (y- 32 )= (x- 4) square trinomial in factored form, (x- 4)2. Presently, the " 16" is 2

1 being added to -- Y . To see the vertical transition, students

2 1 1

should factor - - from the left side of the equation, resulting in -- (y- 32) . 2 2

From this form they can now state that the maximum value is 32 metres, and it occurs four seconds after the projectile begins its flight (refer back to discussion of C9 on p . 60).

vertex -7 ( 4, 32)

Bl!C32 Students might want to enter this equation into their calculators. This requires standard form, so they would multiply both sides by (-2) (inverse operation), then add

32 to both sides (inverse operation). Students should interpret the transformations in this

1 form. The negative indicates a reflection in the x-axis. The " 2 " means a vertical stretch of2.

The (x- 4) means an horizontal translation of +4, and the "+32" , means a vertical translation of

+32.


Quadratics


Pencil and Paper ( C9 /B I I C32)

1) Match the following equations with their graphs and explain how you matched

them. a) y = 0.5.x2- 3x + 4.5 b) y = -3x2 + 1

1 2 2 4 c) y =--x --x--

3 3 3 d) y = x2 + 4x + 9

2 10

-3 3

10

-1

2 3

Pencil and Paper (C9/BI)

2) Explain the student's thinking from step 2 to step 3 . Was the student correct? If so, explain why. If not, explain what they should have done.

y = -3x2 + 12x - 7 ... step 1

y + 7 = -3 ( x 2 - 4x) ... step 2

y+ 7 = -3(x 2 -4x+4)-4 .. . step 3


3) An environmental group wants to designate an area as a natural habitat. One side of the rectangular-shaped area is adjacent to a large lake and does not require fencing. The group is allowed to fence the remaining three sides using 48 krn of fencing. What dimensions will ensure a maximum area for this natural habitat? What is the maximum area?

4) A rectangular playing field is fenced into three sections as shown. If the total amount of fencing used is 800 m, what are the values of x andy so that the area is a maximum?

y


y y

QUADRATICS

Suggested Resources

61

QUADRATICS

Quadratics

Outcomes


will be expected to

C22 solve quadratic equations

B I 0 derive and apply the

quadratic formula

B I demonstrate an



between the arithmetic

operations and the operations

used when solving equations

62

Elaboration - Instructional Strategies/Suggestions

C22 In a previous course, students related the factors of quadratic expressions with

the x-intercepts of the graphs of their corresponding equations to understand that the

solutions of a quadratic equation could be found by factoring. Students also learned to

trace a graph using technology to solve simple quadratic equations that could not be

factored.

C22/BIO/BI In this course, students will study a method that can be used to solve all

quadratic equations. When factoring is difficult, students can use a formula to solve

the quadratic equation. They should derive the quadratic formula and understand

why it can be used to solve any quadratic equation.

0 Teachers should ask students to solve the equation :>?- - 16 = 0 , then :>?- - 15 = 0 , then :>?- + 4x + 4 = 0. For the first one, students will add 16 to both sides and solve

:>?- = 16 to get x = ±4 . For the second, they will use the same procedure and get

x = ±.Jl5. They will need to factor the third one: (x + 2)2 = 0, ultimately, x = - 2 .

Students may derive the quadratic formula by

solving the general quadratic equation

ax2 + bx + c = 0 a c1- 0. Ask students to solve a quadratic equation where the a value is greater

than 1. By dividing by the coefficient of :>?-, students can simplifY the equation and then use

"completing the square." From their experience

with tiles, students should realize how to obtain

the constant term that will produce a perfect

square trinomial. Then, by factoring, finding the

square root, and isolating the variable, students

will solve the equation.

2 b c x +- x=--a a

2 b b2 c b2

x +-x+-=--+--a 4a 2 a 4a2

(x+ :J2

P~~ac b ±.Jb2 - 4ac x+-=-----

2a 2a

-b±.Jb2 - 4ac x =

2a

Ask students to solve the general quadratic equation. The result represents the

solution to any quadratic equation and so can be used as a formula by students when

needed. They may need some practice identifYing the value for a, b, and c.

Students should solve problems that require the use of the quadratic formula. Many

students will use the formula all the time, others will continue to use the factoring

method when the expression is factorable. Still others will use technology and the

features that are available to find the solutions.

Often one of the two solutions does not fit within the context of the problem. This

solution is called an inadmissable root.

ATLANTI C CANADA MATHEMATICS CURRICU LUM MATHE MATICS 3204/3205

Quadratics


Peiformance ( C22/B 1 0/B 1)

I) A football is kicked into the air and follows the path h = -4.x2 + 20x, where xis the time 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 after 6 seconds? d) How long does it take the ball to reach a height of 15 m?

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

3) A rectangular office building surrounded with a uniformly wide parking strip is to be built on a rectangular lot of 4125 m 2

• The building and the parking strip together, when completed, will measure 40.0 m by 60.0 m and take up 4/5 of the building lot. What will be the width of the parking strip?

4) Is it possible to bend a wire 15 em in length to form the legs of a right triangle that has an area of30 cm 2? If so, find the length of each side.

5) The following students were practising deriving the quadratic formula. Examine their attempts, locate all errors, and suggest why they may have made those error(s).

Nate

mx2 +nx+d = 0

2 n d x +-x+-=0

m m 2

x 2 +~x = _:!__+ (~) m m m

( x<J -dm:;n2m x + : = ~ ) m ( n

2 - dm)

n+)m(n2

-dm) x=

m m

}oumaf(C22/B 10)

Kate

px 2 +qx+ 4 =0

x 2 +.f!_ x = _!...._ q q

x2+1.x+2.[P )

2

= -r +2.[P2J

q 2 q q 2 q2

[x +.£.]2 = -r (2q )+ p2

2q 2q2

p ±~p2 -2rq X+- = ----'-=---

2q .J2q

-p±~p2 -2rq X=---'-- ..!...:....-- ...:....

2q

6) Pretend that you are on the phone with a friend who missed the class about how the quadratic formula was developed, and what it is used for. Explain to your friend from where this formula comes, and how and when to apply it.


QUADRATICS

Suggested Resources

63

QUADRATICS

Quadratics

Outcomes


will be expected to

B 11 (Adv) analyse the

quadratic formula to connect

its components to the graphs

of quadratic functions

C22 solve quadratic equations

B 1 demonstrate an



between arithmetic operations

and the operations used when

solving equations

64


B11 (Adv) Students should explore the quadratic formula to help them understand

the ever-present symmetry in the graph of the quadratic equation. For example,

--b ± ..Jb

2 -4ac

students can express the formula in this form x = which should

imply to them that the equation of the axis of

b symmetry passes through -- and that the roots

2a

are removed from the axis of symmetry by this

a1 ±..Jb2 -4ac

v ue . 2a

This, along with the transformational form

1 ( b2

- 4acJ ( b )2

-:; Y + 4a = x + la might be useful to

2a 2a

axis of symmetry, x = 2~ y \

I

b2 - 4ac .,J b2

- 4ac 2a 2a

students who might want to express the vertex of any quadratic equation as

(-!..._, P -4acJ.

2a 4a

C22 As students continue to solve quadratic

equations using various methods, they should be

encouraged to remember that the solutions to the

equations ax! + bx + c = 0 a "# 0 are the x-intercepts

on the related graph. From previous work, students

will understand that the roots, factors, and zeros are

connected. For example, if they solve the quadratic

~ - 3x- 10 = 0 by factoring, they would get factors

of (x- 5) (x + 2) = 0. Since the product is zero,

then each factor may be zero giving roots of 5 and -2. Looking at the graph of

j(x) = ~- 3x- 10, students should see that the x-intercepts are 5 and -2. To solve

~ - 3x- 10 = -2 using the graph, they could translate the image of the x-axis down

two units and read the new intercepts. They should realize they are solvingf(x) = -2.

Students should understand that, when graphing functions, the solution to the related

equation, when equal to zero, can be read from x-intercepts values. This is because the

given function intersects the line y = 0 (the x-axis) at these points on the graph. When

using the graph to solve the equation ~- 3x- 10 = -12, students could graph

y = ~ - 3x- 10 and examine where it intersects the line y = -12. They could read the

horizontal intercepts as x = 1 and x = 2.

ATLANTIC CANADA MATHEMATICS CURR ICULUM MATHEMATICS 3204/3205

Quadratics


Pencil and Paper ( C22/B 1)

1) Explain which method would be best to use to solve each of the following quadratic equations. Then use the method, and one other, to determine the solution(s) for each. Which equation has roots whose difference is the greatest? a) 6.x2 - x - 2 = 0 b) x2 - 4x - 5 = 3 c) 7x2 = 1 - 2x d) 14.x2 = 5x

Performance (C22/B1)

2) Mary, David, and Ron are students in group 1. They have been solving quadratic equations together. Instead of co-operating, each has solved the equation on their own. Here are their solutions. a) Locate any errors, explain why they are errors, and decide which solution

attempt is the best. The equation A = x2 + 3x- 110 represents the area of a field. Students are asked to find the width (x) in metres if the area is 100 m 2

•

Mary

100 = x 2 + 3x -110

x 2 +3x-210=0

X 3±J9-4(1)(-210)

2

3±.r-B31 2

{3+~}

Journal (B 11 (Adv))

David

x 2 + 3x- II 0 = I 00

x 2 +x-10=0

(x -5)(x+3)=0

X = 5,-3

{5,-3}

Ron

x 2 + 3x- II 0 = I 00

(x+55)(x -2)=100

x =-55 or 2 {-55. 2}

3) Pretend that you are on the phone with a friend who missed the class about how the quadratic formula helps explain the symmetry in the graph of a quadratic equation. Explain to your friend about this symmetry aspect.

b 4) Why does -- determine the x-reading of the vertex?

2a

5) a) Sharon was telling Anthony that she could find the x-coordinate for the vertex of the quadratic function y = 5.x2 + 13x- 21 mentally. Sharon said it would be 13 divided by 5. Was she correct? Explain how you know.

b) What would Sharon have to do next to get they-coordinate for the vertex?

journal (C22)

6) Explain why it is important to make a quadratic equation equal zero before trying to solve it by factoring. Is this also true when trying to solve it by using the quadratic formula? Explain how you know you are correct.

7) Explain how you would use a graph to solve this quadratic equation: 3x2- 12x- 7 = -12.25


QUADRATICS

Suggested Resources

65

QUADRATICS

Quadratics

Outcomes


wi!f be expected to

A4 demonstrate an

understanding of the nature of

the roots of quadratic

equations

C 15 relate the nature of the

roots of quadratic equations

and the x-intercepts of the

graphs of the corresponding

functions

A9 represent non-real roots of

quadratic equations as

complex numbers

A3 demonstrate an

understanding of the role of

irrational numbers in

applications

66

Elaboration -Instructional Strategies/Suggestions A4/Cl5 When students solve the equation 4x-2 - 12x + 9 = 0 using the quadratic formula, they will get only one value for x. This value "i.. " is called a double root (two equal real

2

roots). When solving by factoring they would ge{ x - %-J ( x - %-J = 0 . So, x =; and ~.

Students should note that whenever the discriminant ( b2- 4ac) is zero a double root will

result, and that it shows as a single x-intercept on the graph (the graph is tangent to the x-axis at that point).

Students need to explore what happens when a parabola does not intersect the horizontal axis. Ask students to continue to explore using the equation fix) = x2- 3x- 10. Have students solve the equation when fix)= -14 using both the graph and the quadratic formula. The quadratic formula produces a negative value for the discriminant, and the graph does not intersect the horizontal line y = -14. After exploring more examples, students should conclude that there are no real roots when the discriminant is negative.

A9 When the discriminant is negative, there are two imaginary roots, or two roots in the complex number system. To find the roots of fix)= -14 using the quadratic formula, the equation must be rewritten as x2- 3x + 4 = 0. Students would find the discriminant is negative and thus there are two complex roots. The roots are 3 ±H. The His written

h · .z Th. 3±iJ7 2 i-.../ f SinCe l = -1. IS X = . 2

A4/C 15 In summary then, students can have a better sense about the roots of a quadratic by examining the value of the discriminant. When positive, there will be two real different roots, when zero, there will be a double root (two equal real roots), and when negative, there will be no real roots, but two complex roots (one being the conjugate of the other).

A3 Since irrational numbers arise when using the quadratic formula, discussion should centre around whether an exact or an approximate solution is appropriate. Students will recognize that irrational numbers can be written in exact form or by decimal approximations. In applications based on measurements (area and trajectory problems), students will understand that approximations are, in fact, desirable. However, if exact answers are called for, students should be able to express them in simplified form . The simplified form may be more meaningful for comparison purposes. (If one side of a rectangle is .J2, and the other side Jl8 , then, since .Jl8 = 3.J2, students can describe one side as being three times the other.)

A4/Cl5 Advanced-level students should benefit from discovering that if r1 and r

2 are the

-b two roots of a quadratic equation ax2 + bx + c = 0 , then (!j + rz ) = - , and(r. + r.) = .:...

a ' 2 a They may find this helpful when looking for an equation. For, example, if the roots of an equation are 3 and -4, (3 + (-4)) = (-1) , so, -bla = -1, sob/a= 1

c and ( (3) (-4)) = -12 , so,-= -12

a So, the equation could be x2 + x- 12 = 0, or 2xl + 2x- 24 = 0, and so on.


Quadratics


Pencil and Paper (A41 A9)

1) a) If one of the roots of a quadratic equation is 4i, what is the other root? b) State a quadratic equation that has these two roots. Describe the appearance of

its graph.

2) If two roots of a quadratic equation are 112 and 2/3, what might the equation be? 3) Solve for x:

a) .fi x 2 - x- 3.fi = 0

b) J2 x2 = .J3 x+.Ji

Pencil and Paper (A4/A3) (z-level only)

4) Triangle PQR has vertices P (-4, 1), Q (2, -1), and R (1, k). Find all possible

values of k such that .o.PQR is isosceles.

5) For what values oft does :x! + tx + t + 3 = 0 have one real root? 6) Prove that if the quadratic equation p>r? + (2p + 1)x + p = 0 has equal roots, then

4p + 1 = 0. 7) Prove that there is no real value of k for which the quadratic equation

:x!- 10 = k(x- 2) has equal roots. 8) Prove that for all real values of m, the given quadratic equation has no equal roots

(m + 1)r + 2m (2x + 1) = 1 9) Given the general quadratic equation a>r? + bx + c = 0, what relationship must be

true among the coefficients a, b, and c for the equation to have a) two distinct real roots? b) two equal real roots? c) no real roots?

Performance (A4/A9)

1 0) When Chantal was asked to describe the roots of the equation 14r- 5x = - 5, she rearranged the equation so that it would equal zero, then used the quadratic formula to find the roots (see her work below left). Edward said that she didn't

have to do all that, and he then showed the class his work. (See below right.) Are they both correct? Which method do you prefer? Explain.

Chantal

5± --}25 -280 x= ---'--- -

28

5 ± -J-255 x= - ---=-- -

28 so, there are no real roots

journal (A4/CI5)

Edward

b 2 - 4ac

= 25- 280

= -255

:. no real roots

11) Create quadratic equations that have two distinct roots, two equal roots, and no real roots. Explain how you know.

journal (A41 A9)

12) Explain why you know that if a quadratic equation has one imaginary root, there has to be another.

ATLANTIC CANADA MATH EMATICS CURRICULUM . MATH EMATICS 3204/3205

QUADRATICS

Suggested Resources

67

QUADRATICS


Unit 2 (5 °/o)

Rate of Change

RATE OF CHANGE

Rate of Change

Outcomes


wilt be expected to

CI7 demonstrate an

understanding of the concept

of rate of change in a variety

of situations

B4 calculate average rates of

change

70


CI7 Students will explore in a variety of practical situations such as trajectory, travel,

and economic and population growth. They should begin with simple models

represented by linear functions whose average rate of change is constant and progress

to variable rate of change, which implies some sort of curve. Students should be able

to discern different rates of change visually (e.g.,) on a parabola as the independent

value increases and the dependent value approaches the maximum, the rate of change

of the dependent value decreases as the independent approaches the x-coordinate of

the vertex.

CI7 /B4 Students should explore some situations like heart rate, the rate at which

runs are scored in a ball game, birth rate, population growth rate, and employment

rate. They will need to calculate these rates . For example, heart rate can be calculated

by counting the number of beats in a time interval, divided by the time interval.

Heart rate = Number of pulse beats in a time interval

Time interval

Rates are important because they tell students how one thing is changing in relation

to another. For example, the rate at which runs are scored in a baseball game can be

thought of as

Change in the number of runs scored

Change in the number of innings played

Students might read about an oil spill in the newspaper and how pollution control

authorities monitor the spread by estimating the area covered by the oil at different

times, then calculating the rate of spread as

Change in area

Change in time

This would tell them how fast the area of the oil slick is changing.

ATLANTIC CANADA MATHEMATICS CURRICULUM . MATHEMATICS 3204/3205

Rate of Change


Pencil and Paper (CI7/B4)

1) The high school girls basketball team is selling T-shirts to raise money for the season. They set the price at $20 a shirt, but were selling an average of only 60 per week. By experimenting, they found on average, for each extra $5 they charge, they sell 1 0 fewer T-shirts. a) If they increase the price from $20 to $25, how does the number sold change?

What is the average rate of change of sales with respect to price? b) If they increase the price from $25 to $30, what is the average rate of change of

sales with respect to price? c) Explain your findings in a) and b)

Pencil and Paper (B4)

2) David took his pulse when he was resting and counted 20 beats in 15 seconds. Then he did 10 minutes of vigorous exercise and took his pulse again. This time he counted 52 beats in 15 seconds. How much did his heart rate increase as a result of the exercise?

}oumal(C I71B4)

3) If a linear function is defined by y = mx + c, a) What is its rate of change? b) What is the significance of a negative rate of change?

4) Describe how you think statisticians calculate the birth rate in a country. 5) In many sports, data are collected to help coaches and athletes improve

performance. Very often they are reported as rates. Choose a sport you are interested in and see if you can find any rates that are used in analysing performance.


RATE OF CHANGE

Suggested Resources

Barnes, Mary, "Investigating

Change," Unit 1, Curriculum

Corp., 1992.

71

RATE OF CHANGE

Rate of Change

Outcomes


will be expected to

Cl7 demonstrate an



of situations

Cl6 demonstrate an

understanding that slope

depicts rate of change

C30 describe and apply rates

of change by analysing

graphs, equations, and

descriptions of linear and

quadratic functions


change

72

Elaboration -Instructional Strategies/Suggestions Cl7 /Cl6/C30 Students should be able to look at a graph and tell how fast the dependent is changing with respect to the independent value and determine if it is changing at the same rate all the time, or if the rate of change varies over time. If a function is given by y =

j(x}, then students should talk about "the average of ywith respect to x". This would mean the change in y per unit change in x.

Average rate of change of ywith respect to x

Continuing, if x changes from a to b then j(x} changes from f(a) to j(b). In this case, the change in x is (b- a} and the change in y is f (b)-f (a). Iff is linear, the rate of change between a and b, would be the same no matter where a and b were located along the x-axis. They should conclude that the rates of change were constant along a straight line. Iff is curved, then students should expect a variation in the rate of change for different intervals along the curve.

change in y

change inx

y

a b

y

r X

a b

B4 The average rate of change of y with respect to x over the interval a to b, is change iny f(b) - f(a)

change in x b - a

I For example, if b = 5 and a= 2, and/ (x) = -x + 3, then

2

f(b)-J(a)~ [±(2)+3)-[~(5)+3) b-a 5-2

-5 .5-4 3

1.5

3 1 = 0.5or-

2

Some students may choose the points in a different order and, thus , express the formula in a different format. The same answer will result.

J(a)- f(b) ~ [ ~(2)+3 )-[ ~(5)+3) a-b 2-5

4-5.5 -3

-1.5 = --=3

1 = 0.5or-2

X


Rate of Change


Pencil and Paper (C17 /C16/C30/B4)

1) When tested for diabetes an individual was asked to consume a sweetened drink. The blood sugar level decreased slowly as time passed. We can say that blood sugar concentration Cis a function of rime t and write C = J(t). Write an expression for the average of a person's blood sugar concentration over the period from one hour to three hours after they have stopped drinking.

2) You are travelling in an air balloon for the first time. As you increase your height above sea level, the air temperature changes. Suppose that the relationship berween air temperature PC and height above sea level h metres is given by T = J(h). a) Use function notation to write an expression for the rate of change of

temperature with respect to height for the interval from h = 200 to h = 500. b) In dear air, the air temperature decreases by about 1 °C for each 100 metres

above sea level. If the temperature at sea level is 25°C, find an algebraic formula

for J(h) . c) What is the rate of change of T with respect to h experienced by a hot air

balloonist as he rises from 200 m to 500 m? d) Would a glider pilot experience the same rate of change of temperature as she

climbs from 1000 m to 2000 m?

Pencil and Paper (C30/B4)

3) a) Describe in words the average rate of change depicted in the following graphs. b) Calculate the average rate of change in each.

i)

-2

b) interval: x = 2 to

X=3

y = 2x2 - 5x -6

Joumal(C30/B4)

ii)

b) interval: x = -5 to

X= -3

y = -x2- 4x + 7

4) a) Describe what Naomi is doing in the first step of the calculation for the average rate of change, and why.

b) Has she made any errors in finding the average rate of change? If so, correct them.

for y = 0.3x2

from x = 1 to x = 2 2 ?

f(b)- f(a) 0.3(2) -0.3(It ---'-~----'-:__ step 1

b-a 2-1

=-8.8-(-4.7)

=4.1

so the rate of change is 4.1


RATE OF CHANGE

Suggested Resources

73

RATE OF CHANGE

Rate ofChange

Outcomes


will be expected to


change

CI7 demonstrate an



of situations

CI6 demonstrate an

understanding that slope

depicts rate of change

74


B4/CI7 /CI6 Student should understand that finding the rate

of change of a linear function is very easy, because it is the same

over any interval.

The graph of the function is a straight line, and the rate of

change in y change is ). ust the slope of the line, ch · . ange m x

If the function is given by y = mx + b, its rate of change is m.

y

c -t---t----t---7 X

a b

fig . 1

Students should be able to explain that a positive slope indicates a positive rate of

change. Where rate of change of a function is positive, the function is said to be

increasing. For example, if you work longer hours you earn more money if you are

being paid at an hourly rate. A negative slope would indicate a negative rate of

change. For example, as the height above sea level increases, air temperature decreases.

The linear function that has a negative rate of change is called a decreasing function. If

a function has zero slope, there would be a zero rate of change. For example, as the

number of hours of work increase, the amount of pay remains the same, since the pay

is a flat rate instead of an hourly rate. The linear function with a zero rate of change is

called a constant function.

y y

y = f(x)

-t--,-----,---7x

fig 2

For nonlinear functions, the rate of change is not

the same everywhere. Motion is represented often

as a curve. If the graph slopes upward to the

right, it means the rate of change is positive. For

example, for each second that passes as a soccer

ball is kicked into the air, its height continues to

increase until the ball reaches its maximum. At

the maximum, the rate of change is zero. As the

ball begins to fall to the ground, the rate of

change is negative. Where the rate of change of a

function is negative, it is said to be decreasing.

a

.... ..c Ol

"iii I

fig. 3

14

10

2

b

a 1 b 2 3 a' b' 4

Time (sec)

ATLANTIC CANADA MATHEMATICS CURRICU LU M : MATHEMATICS 3204/3205

Rate of Change


Performance (B4/ C 17 I C 16)

1) Phuong has a casual job in which she earns $10.60 per hour. If she works more than 8 hours in a day, she is paid "time and a half," that is, $15.90 per hour, for every hour over 8 hours. And if she is required to work more than 11 hours in a day, she is paid "double time," that is, $21.20 for every additional hour. a) Draw a graph to show Phuong's earnings as a function of the number of hours

she has worked. b) Explain how the graph helps you to tell when Phuong is earning money at the

fastest rate. c) Explain why the graph appears to change steepness after 8 hours.

2) Kim had a job last Saturday. His employer paid him $50 to cut his lawn. Five hours later Kim was finished and walked home. On his way home the mower sprung a leak and left a trail of gas. The tank was leaking 0. 1 litres per minute, and the cost of gas is 53.9 cents a litre. If it took him one hour to get home, draw a graph that shows the relationship between time and net income for Kim's mowing.

3) Harry thinks that when finding the average rate of change there is no difference whether the function is linear or nonlinear. Would you agree with Harry, or not? Explain your chinking.


RATE OF CHANGE

Suggested Resources

75

RATE OF CHANGE

Rate of Change

Outcomes


will be expected to

CI7 demonstrate an



of situations


of change by analysing graphs,

equations, and descriptions of

linear and quadratic functions


change

76


CI7 /C30 Students should conduct experiments and gather data that, when organized

and graphed, will result in situations where the rate of growth or decay changes

between each set of points. The graphs of these situations will result in curved

relationships.

For example, in the rectangular campsite problem (in the quadratic unit, p. 48), the

campers had 50 metres of string to place along three edges of a rectangular camp lot

to maximize the area. From their table of collected values, students observed that,

even though they used widths that increased at a constant rate, the area of the

campsite increased until it reached the maximum, but at a varying rate. The graph

that resulted had a parabolic shape, concave downward. Students should also be asked

to observe when the rate of change was zero. A connection should be made between a

rate of change of zero and the maximum or minimum value for a function.

B4 Some students were asked to calculate the average rate of change for the area of

the campsite when the width changed from 5 m to 6 m, A ( x) = - 2x 2 + 50x. Their

work would look like this:

f(b) - f(a)

b - a

/(6)- /(5)

6-5 228-200

=28m2 of area per 1 m change in width

Others calculated the rate of change for the area in the interval from 9 m to 10 m, and

still others in the interval 11m to 12 m .

from 9 ---7 10 ==> f(IO) - f( 9) ==> 300 - 288 = 12m2 per 1 m change in width. 10-9 1

from 11 ----" 12 /(12)- /(11) 312-308 4 2 1 h . "d h ---- ==> ==> = m per m c ange m WI t . 12-11

C30 Clearly, students can see that the rate of change is different on different intervals,

which is typical of nonlinear situations.


Rate of Change


Peiformance (Cl7 /C30/B4)

1) Ask students to go back and revisit a problem (see reference below) worked on in the quadratic unit. Construct a graph to represent the situation and in each discuss how the average rate of change (of the phenomena studied) changes: e.g., increasing, decreasing, speed, etc. at various intervals along the curve. a) kicking a football-hangtime problem (p. 61) b) the campsite problem (p. 46) c) Chantal's bathtub problem (p. 51) d) the golf shot problem (p. 49)

2) Again, referring to the above question, calculate the average rare of change for each using the following information: a) Find the average rate of change of the height of the ball after it has been in the

air between two and three seconds. b) Explain how you would use the average rate of change in the area to determine

the dimensions that result in the campsite having a maximum area.

For parts c) and d), create a problem you would ask your partner to see if he/she understands average rate of change.


RATE OF CHANGE

Suggested Resources

77

RATE OF CHANGE

Rate of Change

Outcomes


wifl be expected to


instantaneous rate of change


of change by analysing graphs,

equations and descriptions of

linear and quadratic functions

CIS demonstrate an

understanding that the slope

of a line tangent to a curve at

a point is the instantaneous

rate of change of the curve at

the point of tangency

C27 approximate and

interpret slopes of tangents to

curves at various points on

the curves, with and without

technology

C17 demonstrate an



of situations

78

Elaboration -Instructional Strategies/Suggestions C28/C30 People often talk about the rate at which the population is increasing or decreasing. This rate can be found by determining the change in population over a time interval and dividing by the change in time. For example if the population of the nation increased by 2.4 million over a six-year period, students could say that the average rate of increase over six years is 0.4 million per year. Sometimes it makes sense to talk about rates of change over very short periods; for example, credit card interest rates, heart beats per five seconds, typing words per minute, car speed in kilometres per hour, speed of sound in metres per second. By analogy, it should make sense that invested money can earn interest as fast as it can be calculated (so much per second). Students should agree that it does make sense to talk about someone's speed at any instant, even if that speed is changing. For example, when a student rides a bicycle or drives a car, the speedometer shows the speed at every moment.

C28/C18/C27/Cl7 Students can find the average rate of change of a function from its graph. The slope of the straight line joining the two points that represent the interval is the average rate of change over that interval. What if those two points come from an interval across which the graph is curved? For example, the average rate of change off from x = a to x = b is equal to the slope of the secant line PQ. This is

y

a b

not the same thing as the instantaneous rate of change at P or at Q but, instead, is the average rate of change across the interval from P to Q.

Students should then imagine what will happen if the graph is "de-zoomed" to its original size. The points Pand Qwill be very close together, in fact, difficult to distinguish. As "a" gets infinitely close to "b" the line PQ appears to be tangent to the curve at Q e.g. , it appears to be touching at only one point. This slope would approximate the instantaneous rate of change at the point Q on the curve. Students can increase the accuracy by taking the two points closer, and closer to Q. For example, usingf(x) = XZ and letting Q be located at x = 2 (b),andselectingPatx= 1.75 (a) , then

j (b) - j (a ) => / (2) - /(1.75) => 4-3.06 => 3

.7

b-a 2 -1.75 0 .25

Next select P (1 .99) to be very close to Q. Then /(2)- j(l.99) 4-3.9601 0.0399

==> ==> ==> ==> 3.99. 0.01 0.01 0.01

From this, students should understand that as "a" gets closer and closer to "b", they will be able to estimate the slope of the tangent line at Q to be 4. In fact, the graphing calculator algorithm uses a similar process where the slope at a point (a,j(a)) is

f(a+0.0001)- f(a-0.0001)

(a +0.0001 )-(a -0.0001)

continued ...

ATLANTIC CANADA MATHEMATICS CURRICULU M: MATHEMATICS 3204/3205

Rate of Change


Performance (C28/C30/Cl8/C27C l7)

1) A projectile was propelled upward into the air from a height of 15m at a velocity of 25 m/ sec. What was its speed at the instant it hit the ground? a) Recall that acceleration due to gravity is approximately -9.8m/sec2

; in free fall the height is -1/2 gfl. Use this fact and the model h = -112 gf2 +v,.t + H where v i

is initial velocity and His initial height, to write the function for this situation. b) Graph the function using a graphing calculator.

i) When did the projectile reach its maximum height? ii) What was the maximum height? iii) How many seconds was the projectile in the air?

c) Recall that average velocity is change in height divided by change in time. i) During what time periods did the projectile have positive velocity? ii) When did it have negative velocity? iii) Was the velocity ever zero? Explain. iv) Find the average velocity from the moment of projection to the moment

the projectile reaches maximum height. v) Approximate the velocity at time 2 seconds; at time 4 seconds.

d) Find the following average velocities, and from those predict the instantaneous velocity of the projectile at 5.5 seconds: i) between t = 3 and t = 5.5 seconds ii) between t = 4 and t = 5.5 seconds iii) between t = 5 and t = 5.5 seconds iv) between t = 5.25 and t = 5.5 seconds

e) Solve the original problem, "What was its velocity at the instant it hit the ground?"


RATE OF CHANGE

Suggested Resources

Meiring, Steven P. et al. A Core Curriculum: Making

Mathematics Count for

Everyone Addenda Series

Grades 9-12. Reston, VA:

NCTM, 1992.

79

RATE OF CHANGE

Rate of Change

Outcomes


will be expected to

CIS demonstrate an

understanding that the slope

of a line tangent to a curve (at

a point) is the instantaneous

rate of change of the curve at

the point of tangency

C17 demonstrate an



of situations

C27 approximate and

interpret the slopes of

tangents to curves at vanous

points on the curves, with

and without technology

80


C18/C17 The previous pages describe how students should come to understand that

an instantaneous rate of change can be approximated by examining average rates of

change (slopes of secant lines) over smaller and smaller intervals. That is, the

instantaneous rate of chagne at Q is approximated by the slope of PQ. As P moves

f(b)- f(a) close and closer to Q, they should recognize that a calculation of

b-a

will generate values for the average rate of change, and that by choosing values for

"a" progressively closer to "b"the resulting values get closer and closer to the

instantaneous rate of change. The instantaneous rate of change then would be the

slope of the tangent line at that point on the curve.

C27 Technology allows students to confirm their estimates for instantaneous rates of

change. It should be noted that since students have not studied limits in this course,

they do not have a method for calculating the exact value of the instantaneous rate of

change. This will be developed in a later course.

The table feature on the graphing calculator can be used to facilitate hand calculations

for the approximate value of the instantaneous rate of change at points on a curve. For

example, to find the slope of the tangent line at x = 3 on j(x) = >?, students would

enter y = >? at ~ = on the function screen. They would enter the expression

(yl(3)- JJ(x)) (} _ x) at .Y; = and explore the table for y values as x approaches 3, 3.1, 3.01,

3.001.

From this they could approximate the slope of the tangent line at x = 3 to be 6. They

would interpret this to mean that the function y = >? has an instantaneous rate of

change of 6 at x = 3.

Once students demonstrate understanding for the instantaneous rate of change at any

point on a curve being the slope of the tangent line at that point on the curve, they

might want to use the graphing calculator to add tangent lines to the functions at

specific points, and to approximate the instantaneous rate of change at those points.

For example, by selecting 5: tangent (in the Draw menu of the TI-83 or TI-83+,

typing in the x-value for the point of tangency, and pressing Enter, students will see

the tangent line drawn to the curve. The technology records, at the bottom of the

screen, the x-value of the point of tangency and the equation for the tangent line (in y = mx + b form). Knowing that the slope of the tangent line is the instantaneous rate

of change at that point, students can read its approximate value from the given

equation.


Rate of Change

Worthwhile Tasks for Instruction and/ or Assess ment

Peiformance (CI8/CI7/C27)

1) In order to determine a projectile's displacement h from the ground, in metres, at any particular time, t, in seconds, the formula h = H + v; t- 4 .9f' is used, where v; is the initial velocity, in metres per second, and His the initial height, in metres, from which the projectile is launched.

v~ IH lh

I

hitting high fly balls. d and hit the ball at a

a) Babe Ruth, the all-time Yankee slugger, was renowned for One day he made contact with a ball 1 m above the groun speed of29.4 m/s. State the equation. Determine the ave the height of the ball during the interval between 2 and 2

rage rate of change of . 5 seconds. Determine

the maximum height of the fly ball. b) How long did the ball remain in the air? c) What was its speed the instant it hit the ground? d) Draw a tangent to the curve at x = 2 and explain how it c an be used to estimate

the speed of the ball 2 seconds after it was hit.

Is according to the 2) An arrow is released with a initial speed of 39.2 m/s. It trave path h (t) = 39.2t + 1.3- 4 .9r, where his the height reached , in metres, and tis the time taken, in seconds. a) Determine where the arrow's speed is the least. Explain. b) Determine the speed of the arrow 1 second before impact

3) The table shows the precipitation data for a tropical region . As the amount of precipitation goes up to a maximum and drops again later in the year, the situation may be modeled with a quadratic function. a) Assuming that the model is reasonable, find the

quadratic function for the amount of precipitation in terms of the number for the month of the year.

b) Extension: Using the equation and what you know about slopes of tangents, produce a series of tangent lines on a graph from which the graph of the equation can be determined.

ATLANTIC CANADA M ATHEM ATICS CURRICULU M: MATHEM ATI CS 3204/3205

Month Precipitation

Jan 5mm

Feb 60mm

Apr 140 mm

Jul 185 mm

Aug 180mm

Nov 105 mm

RATE OF CHANGE

Suggested Resources

81

RATE OF CHANGE

82 ATLANTIC CANADA MATH EMATICS CURRICULUM MATHEMATICS 3204/3205

Unit 3 Exponential Grovvth

(25 - 30 °/o)

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

C2 model real-world

phenomena using exponential

functions

B2 demonstrate an intuitive


recursive nature of

exponential growth

84

Elaboration -Instructional Strategies/Suggestions C2 Exponential functions (like y =air) form another important family of functions . Exponential functions are extremely useful in applications that include population growth, compound interest, radioactive decay, and value depreciation. Consider a very common classroom growth example: The teacher has some important data saved on her graphing calculator. She wants to share it with all32 students. She asks the class to consider which method would be the best to share the data and how much time would be saved (it takes one minute to link and transmit the data).

Method 1: Teacher links with each student and transmits the data one person at a time. Method 2: Teacher links with one student, then each of them links with a student, and so on.

Solution:

Students may try to find a common difference with the table in Method 2. The fact that there is no common difference when sequences of differences between successive terms is examined at D

1 and

02

level signifies that the relationship is not linear or quadratic.

B2 Upon further study of the

Method 1

# of people with

Time period data

0

2

2 3

3 4

3 1 32

32 33

3 5 36

Method 2

# of people with

Time oeriod data

0

1 2

2 4

3 8

4 16

5 32

6 64

pattern in Method 2 (as seen below) students might determine a relationship that they could describe in words as "the number of people (t) with the data during any time period (t) can

Time Period

0

2

3

4

5

6

n

M2 = M 1 • 3

M5 = M4 • 3

# of people be determined by doubling the number in the with data previous time period."

1 = 1 -7 2° Using symbols: t2 = t

1 • 2 , t

3 = t

2 • 2 and so on,

2·2·2 = 8-7 23

2·2·2·2 = 16-7 24

• -7

leading to t . = t . _1• 2, some students will continue looking at the pattern to see that it can be symbolized by 2". Students should state that each term in the sequence is a multiple of each previous term, and that this multiple is the common ratio between consecutive terms (e.g., t. + t._1 ).

Another example might be to consider the following pay schedule: Students might describe this pattern as "You get paid one dollar on day one, then each day after you get three times as much as the previous day. "Thus the amount of money earned (M) on any given day can be determined by tripling the amount earned on the previous day. Using symbols:

and so on, leading to Mn = M. _1

• 3

continued .. .


Exponential Growth


Pencil and Paper ( C2/B2)

1) This is an old story that is found in many similar versions. According to one version, King Shirham of India wanted to reward his Grand Vizier, Sissa Ben Dahir, for inventing and introducing him to the game of chess. The Grand Vizier requested as his reward that the king have one grain of rice placed on the first square of the board, two on the second, four on the third, eight on the fourth, and so on, until the board was filled according to this pattern. The king tried to talk him out of this silliness because he was willing to give him jewels or money, but to everyone's amazement, the Grand Visier stood his ground. Did he do the right thing?

Square

1

2

3 4 5 6

10

25

64

Rice on the Square

1

2 4

Rice on the Board Pattern? Complete the chart. Look for a pattern and describe it. Determine the amount of rice on the 64th square. Determine the amount of rice on the whole board.

2) Imagine you have won a sweepstake, you can chose between a lump sum of $2 000 000 or $10 000/mo for the rest of your life. You decide that in either case, you will deposit half the money in a savings account that earns interest every month at a rate of 0 .5% per month. Investigate to determine which is the better deal? Explain.

journal (B2)

3) Given the two sequences below, explain why one of them will produce terms that depict exponential growth. Explain why the other will not. i) tn =2tn-l +tn-Z' t 1 =5, t 2 =9

ii) t n = O.Stn-l' t 1= 0.25

4) List the first five terms of this sequence:

t 1 =4, t2 =3, t" =3t+2t"_2 , n>2EN

Describe the sequence. Do you think you could predict the 1Oth term? Explain.

5) Steve said thatj(x) = Y+ 2- 2(x + 1) defines an exponential function, when the

domain is x > Oix E R . Is he correct?

... continued


EXPONENTIAL GROWTH

Suggested Resources

85

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

C2 model real-world


functions

B2 demonstrate an intuitive

understanding of the recursive

nature of exponential growth


exponential and logarithmic

equations

Cll describe and translate

between graphical , tabular,

written , and symbolic

representations of exponential

and logarithmic relationships

C4 demonstrate an



and geometric and relate

them to corresponding

functions

86


.. . continued

B2 Students should be able to see that each term in the sequence is triple the

previous term, and could be symbolized using y- 1•

Students might consider how to symbolize getting paid $2 on the first day, then

tripling each day thereafter. As they develop the table to the left, each term is

multiplied by 3 to get the next term leading to the amount paid on the nth day.

day

1

2

3 4

n

Money($)

2

2·3

2 ·3·3

2 ·3 ·3 ·3

2 ·3"-1

It is often easier for students to work with exponential

situations by using the relationship between successive

function values; in fact, their first recourse in evaluating

exponential functions is often through calculating the next

value from the present value.

Sequences where determining the nth term requires

knowledge of all preceding terms are called recursive rules.

The replay capabilities on graphing calculators permit recursive patterns to be easily

evaluated. It does not take long, however, for students to realize that even with

instant-replay capability, recursive patterns do not efficiently determine output for

very large input values. For example, students would need many term values to find

the money earned for day 50 of the money tripling function if they know only the

recursive definition. Once students understand this difficulty, they will be ready to

appreciate that function rules for recursive patterns can be stated in an alternative and

arguably more usable way. However, the sequence editor and table feature on the

graphing calculator can make this much more efficient.

This outcome (B2) will be formalized in a later course.

C2/C25 During their study of exponential growth, students should examine some

issues that have become important in today's society, like world population growth,

ozone layer depletion, pollution control, etc. The tendency of populations to grow at

an exponential rate was pointed out in 1798 by an English economist Thomas

Malthus, in his book An Essay on the Principle of Population. Malthus suggested that

unchecked exponential growth would outstrip the supply of food and other resources

and lead to wars and disease. He wrote "Population, when unchecked, increases in a

geometrical ratio. Subsistence increases only in an arithmetic ratio."

C11/C4 Students should discuss this Thomas Malthus quote and the consequences.

During their discussion they will gain a better understanding of geometric and

arithmetic ratio.

Students should re-establish that functions whose first term is C and whose other

terms are generated by multiplying the preceding term by a constant, a, can be

expressed in the formf(n) = Ca"·', where n,n;:::: l is the number of the term and the

variable a in this situation is the common ratio. By contrast, students should

remember that terms in arithmetic sequence are generated by a linear function where

each term results by adding a constant to the previous term.


Exponential Growth


continued ...

Pencil and Paper (C4)

6) Which of the following sequences are arithmetic, which are power, and which are

exponential?

a) 3, 5, 7, 9, 11, ...

1 1 1 1 b) ~ ' (; '~'54 c) -6,-4,-2,0,2,4, 6 d) -2.988, -1.963, -0.8889, 0 .3333, 2, 5 e) 8, 4, 0, -4, -8 f) 2,4, 8,32 g) 0.01235, 0.03704, 0.1111 ' 0.3333, 1' 3

1 1 1 1 h) ~'(;'l()'J:i

i) .J2,2,J8 j) 4, 5, 12, 25, 44

Activity (C2/C25/C4)

7) Ask students to read the following excerpt from the book The Origin of Species by Charles Darwin, published in 1859. It discusses exponential growth. ''A struggle for existence inevitably follows from the high rate at which all organic beings tend to increase. Every being, which during its natural lifetime produces eggs or seeds, must suffer destruction during some period of its life, ... otherwise, on the principle of geometrical increase, its numbers would quickly become so inordinately great that no country could support the product. Hence, as more individuals are produced than can possibly survive, there must in every case be a struggle for existence ... It is the doctrine of Malthus applied with manifold force to the whole animal and vegetable kingdoms ... There is no exception to the rule that every organic being naturally increases at so high a rate, that if not destroyed, the earth would soon be covered by the progeny of a single pair." Ask students to discuss this in their groups and respond to "Do you think the ideas are still relevant?"

Performance (C2/C25/Cll/C4)

8) A certain bacteria have a doubling time of 20 minutes. If you started with a single organism, with a mass of about 10 - '2gm, and it grew exponentially for one day, what would be the total mass produced? What would it be after two days? How does this compare with the mass of the earth (approximately 6 x 10 21 tonnes)? Is this possible? Explain?

9) a) You have two parents, four grandparents, and eight great-grandparents. If you go further back in your family tree, how many ancestors would you have n

generations ago? b) Decide approximately how many years equals one generation and work out

roughly how many of your ancestors would have been alive 2000 years ago. Is this possible?


EXPONENTIAL GROWTH

Suggested Resources

87

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


wif! be expected to


characteristics of exponential

and logarithmic functions




relationships



notation



equations

88

Elaboration -Instructional Strategies/Suggestions C33/C29/A7 Students should examine the characteristics of the curves of exponential graphs. Ask students to graph the relationship explored on p. 84 method 2, e.g., graph y = 2x and explain why the graph (from left to right) curves slowly at first, then much more quickly. Then graph y = 3x and compare its growth rate with they= 2x.

Ask students to use technology to explore the behaviour of these two graphs using the window settings -10::; x::; 2, and range -1::; y::; 10 . Have them describe the curving behaviour and explore the behaviour at the extreme left and extreme right. Have them predict the behaviour if usingx-values in the domain smaller than -10, or y-values greater than 1. Then have students explore the behaviour of these two graphs using the window settings -10::; x::; -1 and -0.1::; y::; 10 . Have them change the range to -0.1::; y::; .01 and the domain to -15::; x ::; -5 . Have students discuss the behaviour of the graph as it approaches the x-axis. Have them explain why the graphs on this screen look so similar to the graph on the previous screen.

Ask the students if they think either of the two graphs will ever intersect the x-axis and to explain their answers.

Ask students to graphy =50(± )~sing 0::; x::; 10, and 0::; y::; 60. Explore whether this

curve intersects the x-axis. Have students confirm that it will not. Ask students why this function is decreasing. Ask students to describe a situation that could be represented by this graph.

This graph could represent the story of the grasshopper jumping towards a fence. It starts 50 metres away and jumps halfWay to the fence with each jump. If it continues to jump halfway to the fence with each jump, will it ever hit rhe fence? The x-axis is a horizontal asymptotea line to which a curve approaches but will never intersect where lxl is very large.

All exponential curves have a horizontal asymptote. Ask students: "In all the explorations above was the x-axis a horizontal asymptote?"

The family of functions used to describe the patterns of change in the wage-tripling situation or the data-transfer situation are called exponential functions. Properties of exponential growth and decay situations can be studied by examining exponential functions of the form j(x) = abx.

C25 Exponential functions have another important property. No matter what the current value of the function is, it takes the same length of time to double or triple in size. By extending the table, students could find the number ofhours it takes for the bacteria to triple, or quadruple, etc. In the table below, students can see that the number of bacteria always doubles in three hours. This property provides a quick way to estimate the future size of a population.

Time in hours 0 1 2 3 4 5 6 7 Number of bacteria 100 126 160 200 250 316 400 500

When numbers are small, doubling does not cause a very big increase, so the graph has a fairly gentle slope. But when the numbers are large, doubling makes an enormous difference, and the graph tends to become very steep. This is the typical shape of a exponential growth curve.


Exponential Growth


Performance (C33/C29/A7 /C25)

1) Use your calculator to draw a graph of the function defined by y = 2 x- 1. Set the

window, 0 :::; x:::; 64 and 0::; y ::; 100 . Describe how your graph looks.

a) Does this graph model the rice on a checker board problem from p. 85? Explain.

b) Note that even though we would like to see what happens all the way up to 64 squares, as soon as x gets to be 5 or 6, the value of y increases very rapidly, almost in a vertical line on a graph. One way to help this problem is to change the range for they-value to something much bigger, say 100 000. Make this change and then redraw the graph of this function .

c) Describe how your graph looks. d) How would you describe the asymptote on this graph?


2) Kareem is a car enthusiast. He owns a new Mercedes, worth $60 000, and a vintage Corvette, worth $30 000. The new Mercedes will appreciate at 5% per year, and the Corvette will appreciate in value at 10% per year because both are collectors' items. a) How long will it take the value of the Mercedes to double? b) How long will it take for the value of the Corvette to double? c) Establish a formula for the value of the Mercedes after t years. d) Establish a formula for the value of the Corvette after t years. e) Will the two cars ever have the same value? If so, when? If not, explain why

not.

Pencil and Paper (C33/C29/A7)

3) Given the following representations of exponential functions, describe the relationship in words, then write the domain and range for each.

ii) y

iii) iii)

D 2 3

1000 1166 40 1259 71 1360 49


EXPONENTIAL GROWTH

Suggested Resources

89

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

C34 demonstrate an

understanding of how


graphs of exponential

functions



equations

90


C34/C25 Consider questions like: "What could I expect to earn on day 22 (given the

wage-tripling scheme described earlier) ifl were to earn $10 on day one?" "Which car

would be more valuable in three years-a $15,000 car that depreciates 30% every year

or a $12,000 car with a 20% depreciation rate?"

Using exponential functions of the form f {x) = Ctr to answer questions like these

amounts to changing the values of C and a. An activity like Exploring the Graphs for

Exponential Functions (see below) engages students in exploring the effects of

changing the values of the parameters C and a on the graph off {x) = Ctr.

In the activity Exploring the Graphs ofExponential Functions (NCTM Addenda

Algebra) students are given equations like j(x) = 0.6 x, j(x) = 4(2.3Y, etc. and asked

to enter them into a graphing utility, look at their graphs and their tables of values,

and to respond to:

1) Describe the trends (growth and decay) in the relation between x andy values. 2) Describe any x-values that result in the same y-value. 3) Describe the overall affect of Con the tables and graphs for j(x) = Cax 4) Consider two exponential functions j(x) = ax and g (x) = if where a and b are

distinct positive numbers. i) What is the effect if a < b and each is greater than 1; less than 1? ii) What is the effect if a> b and each is < 1 less than 1; greater than 1?

From this investigation students should make conjectures about the effect that C and

a have on the functionf(x) = Cax. It is important that they test and verify their

conjectures.


Exponential Growth


Pencil and Paper (C 34) for Advanced-level students

1) Describe what happens to the curve of the graph of the function l(x) = b Ax (where A > 0) as x increases without limit. What happens to the curves as x decreases through the negative real numbers?

2) Describe what happens to the curve of the graph of the function l(x) = b Ax (where A < 0) as x increases without limit. What happens to the curve as x decreases through the negative real numbers?

Performance for Advanced-level students

3) Graph the function l(x) = 8 Ax for A = 1, 0.5, and 0.25. a) Show algebraically how they-coordinate of a point on the graph of I {x) = 8Ax is

related to they-coordinate of the point vertically above or below it on the graph of I {x) = S0·5x.

b) Describe how the graph ofl{x) = 8Ax changes as A approaches 0.

4) Graph the functionl{x) = 8Ax for A= -1 , -0.5, -0.25. Describe how the graph of l(x) = 8Ax changes as A approaches 0. What transformation would map the graph of l(x) = 8Ax onto the graph of l(x) = s-Ax ? Explain.

5) a) How are the graphs of l(x) = 5 x and g (x) = -5 x related? b) How are the graphs ofl(x) = 5 x and h (x) = 5 -x related? c) Do the graphs I and h intersect? If so, at what point(s)?

Pencil and Paper (C34) for Advanced students:

6) Given the function f, defined by l(x) = b Ax where b > 1, A > 0 and where x is real. a) Prove that there is a doubling time, d such that l(x + r1) = 2 l(x) for all x. b) Write an equation that relates A, b, and d and is independent of x.

7) D escribe in words the transformations of y = 3 x andy = 2 x that are visible in the equations given:

i)

ii )

1 ( 3) -3x -- y- =2 2

J + 5 = 32(x-1)

8) D escribe the approximate location and position of graphs of the following eq uatio ns:

i)

i i)

.!.(x+4) -y=Y

.!_(J + 2) = T (x-1) 2

9) Create a problem where this equation would need to be solved for the variable C. 10 000 = C ·2°·5x


EXPONENTIAL GROWTH

Suggested Resources

91

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

C2 model real-world


functions



equations



written and symbolic



92


C2/Cll/C25 Students will investigate typical growth patterns (population growth,

economic growth, etc.) and learn about the exponential functions used to model

them. For example, students might be asked to study graphs that deal with various

types of growth and answer questions like the following:

1) How are they the -~

same? ~ 600 u

2) How are they different? m (l)

400 0 Q5

.D 200 E ::::l z

1 2 3 4 5 6 7 Time (hours)

3) Describe the growth represented here.

Students might be asked to look at tables of values to find

patterns:

Time in hours 0 1 2 3 4 5 6 7 Number 100 126 160 200 250 316 400 500

1) Number ofbacteria present at different times:

Year 1995 1960 1965 1970 1975 1980 1985 1990 Population 2.70 2.98 3.29 3.63 4.01 4.43 4.84 5.29

§ 150 ·.;::; ro :; c. 100 & ....... :g 50 ro

a:

~ 40 ..c: g. 20

.3

2 4 6 810 Time (months)

20 40 60 80 Age (days)

~100~ ~ 500

2 4 6 Age (years)

2) The table gives more detail of population increases from 1955 to 1990. The numbers are in billions.

Students should calculate the ratio of each entry in the second row to the entry in the

previous column. Ask students what they notice.

From these tables that contain collected data, students should notice that the growth

160 126 ratios are more or less constant, e .g., in the first table

126 = 1.26 =

100. In general,

students should learn that if growth ratios are constant, then they can model the

growth with a function in the form j(x) = Ca x, where Cis the size of the starting

population and a is the growth rate. Functions of this type are known as exponential

functions.


Exponential Growth


Peiformance (C2/C25/Cll)

1) The table below shows the actual world population figures according to a census bureau for the period from 1800 to the late 1900s. a) How long did it take the world population to double from 1 billion to 2

billion? Is this more or less than the doubling time associated with the exponential function above?

Year 1801 1925 1959 1974 1986 1997

Population in billions 1 2 3 4 5 6

b) How long did it take the population to double from 2 billion to 4 billion? How close is this to the doubling time computed above? How long will it take to double from 3 billion to 6 billion? Is this more or less than the doubling time predicted by the exponential equation? Is the world population growing exponentially? Is its growth exceeding exponential growth? Explain your answer.

2) If you deposit your money in the bank, you will receive interest on this money. Typically, it is compounded monthly. This means that after the first month goes by you will get interest on the original amount deposited and then the next month you will get interest on the original amount as well as interest on this first interest. This is the principle of compound interest. Normally, when the bank quotes the rate of interest they give it on an annual basis, along with how often it is compounded. For example, 6% per annum compounded monthly. It is important to remember that 6% per annum is 0.5% per month (0.005 in decimal form). a) Complete the table from the headings given, for months 1, 2, 3, 4, 12, 24,

120, 240 and 480, and finally n. b) What is the expression for the amount at the end of the nth month?

Month Expression for the amount at the start of the month

$100

Expression for the amount at the end of the month $100 (1.005)

Amount

$100.50

c) Plot a graph, and from it, determine the amount after the 200th month. d) i) Does your money increase in value quickly at the start?

ii) Does your money increase at the end of the 40 year period? iii) By how much has your money increased over the 40 year period? iv) Think of a situation in which this kind oflong-term financial planning

would be appropriate.


EXPONENTIAL GROWTH

Suggested Resources

93

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

C2 model real-world


functions



written and symbolic



94


C2/Cll As stated on the previous pages, students will investigate typical exponential

patterns. Some exponential patterns show a decay process rather than growth. When

studying the graph of the temperature as a cup of coffee cools and the graphs of the

decay of a radioactive element, students will notice that the graphs are decreasing

throughout the domain and approach a horizontal asymptote as the values in the

domain increase. Ask students to explain what is different about an equation that

results in a graph that depicts a decay process. (See C34.) When studying the graphs

of various types of decay (the decay of a swinging pendulum, the cooling of a cup of

coffee, the decay of a radioactive particle), students should

1) notice how the graphs are alike and how they are different 2) be able to describe the decay represented-how quickly the decay occurs, what is

the half-life, what does the asymptote represent

In their continued study of tables of graphs they should be able to describe all the

characteristics of an exponential relationship, only now as a decay, rather than a

growth process.


Exponential Growth

Worthwhile Tasks for Instruction and/or Assessment Performance (C2/Cll)

1) Carbon 14 is an isotope of carbon found in plants, which have absorbed it from the atmosphere. From the time the carbon 14 is absorbed, it decays exponentially. The percentage, y, of carbon 14 in the plant x years after absorption is given by the following equation:y = 10 2 - 0·00005235x. Graph this equation and determine the half-life of carbon 14.

2) When one face of a cube is painted red and tossed, the probability that it will land 1

redside up is 6 because each cube has six sides, and only one of those sides is painted

red. Tossing many cubes and knowing how many will show red faces is an 1

unpredictable, random process. Rarely will 6 of the cubes do this on any toss.

However, if you repeat the toss many, many times, the average number that show

1 red will approach 6 . So, if one were to toss 100 cubes and remove the red ones,

and continue this, it would take about four tosses for approximately half of the cubes to be removed, so the half-life of a group of cubes is about four tosses. (After

5 1 5 5 25 . one toss 6 remain, 6 has decayed; after two tosses 6 of 6, or

36 remam; and

5 5 5 625 after four tosses 6 of 6 of 6 or

1296 of the cubes are left, 52 have been

removed) .

In this model, assume that the removal of a cube corresponds to the decay of a radioactive nucleus. The chance that a particular radioactive nucleus in a sample of identical nuclei will decay in a second is the same for each second that passes, just as

the chance that a cube would come up red was the same for each toss ( ~) . The

smaller the chance of decay, the longer the half-life (time for half of the sample to decay) of a particular radioactive isotope.

Follow this procedure: a) Toss the 100 wooden cubes onto a table surface. Remove all the cubes that land

red side up. Place to the side. b) Gather up the remaining cubes and toss them again. Again, remove all the cubes

that land red side up. c) Repeat this experiment until there are 10 cubes remaining. (Given the small starting

number, statistically numbers below 10 become insignificant. In actual situations, the number of atoms is much greater.)

d) Preparing to graph: i) What does the number of trials represent in real-life? ii) What does the number of cubes remaining after each toss represent? iii) What is the dependent variable, and what is the independent variable?

e) Use regression analysis to obtain a graph and an equation that best fits the data. f) Determine the half-life from your graph and equation. Show all your work. g) Approximately how many radioactive nucleii will remain after 10 time periods?

Explain. h) According to your model how much time will pass for the 100 nucleii to decay to 15?


EXPONENTIAL GROWTH

Suggested Resources

95

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

A5 demonstrate an

understanding of the role of

real numbers in exponential

and logarithmic expressions

and equations

96


A5 As students model situations and solve problems that can be represented by

functions like y = Ccr· they need to be sure that the expression d' has a meaning for

all values of x. The way cr is defined for rational and irrational values of x is explained

below.

Rational exponents:

Students should already be familiar with the definition of cr in the case where xis a

positive or negative integer.

If m and n are both positive integers,

a 11n =if;;, which is the number whose n'h power is a

amtn =~,which is the number whose n'h power is~

These four definitions give a meaning to cr for any rational number x.

Irrational exponents:

If X is irrational in y = cr' then the value crwill be an approximation. For example,

given y = 2../3, students might choose .J3:::: 1.72, and Y = 21.72

This results in

y = 3.29436 . However if technology is used to evaluate y = 2../3, the result will be

y = 3.322 . In this course, students will not need much emphasis on how cr is defined for irrational values of x. The concept of cr where xis irrational is studied in a

future course.


Exponential Growth


Performance (A 5)

1) a) Apply the definition ofb', where tis rational, to evaluate the following without

3 2 ~ ~ 3

164' 27 3 ' 325, 1253 ' 25-2 I I 2 2

using a calculator: 83 8-3 83 8-3 ' ' ' '

b) Evaluate using your calculator. c) Check using a graphing utility.

2) Which of the following is of greatest value? Explain.

a) (33)3 b) 333 c) 33' d) 333 4

3) Prove that (z3) * 2 7

4) Simplify each expression and evaluate if possible:

a) 42 ·(23 )+32

b) (3xyz)3 ·(9x2yf +(3x)5

(xm)2( Xn+l)2 c) --.. - - -n

X X

5) Simplify: i) 5n + 5n ii) 2X+ 5 + 2X+ 5

6) Explain how 3k + 1 can be the same as 3(3k). Explain how you might use this idea

in a question like " ... factor 1 - 2 2x+ 1 + 42x .•. " .

7) Give the meaning of each power:

1 2 a) x - 5 b) r~ c) a3 d) n-Y

8) Evaluate: 3 - I

a) 164 b) 64 2

d) 2 o .!. ( 1)T 5 +10 -42 -9

9) Explain how you could use a graph to evaluate each of the following:

a) 7o 3333 b) _2

-;

c) 3·2 x- 5 = 2 ·10 -2x-IO


EXPONENTIAL GROWTH

Suggested Resources

97

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

C2 model real-world


functions



collected data



equanons

F 1 create and analyse scatter

plots and determine the

equations for curves of best

fit, using appropriate

technology

98


C2/C25/C3/Fl Students should realize that when they work with real data, there is

always some variability due to measurement error. It is often difficult to measure

exactly and sometimes it is necessary to measure indirectly. Thus students cannot

expect to generate a model that will fit the data perfectly. No matter how accurate the

data, they may have to simplify a complex situation in order to construct a modeL

Assuming the growth ratios are constant is a way of simplifying the situation. When

students think they have found a model, they should always graph it and the raw data,

to see how well it fits. They need to ask themselves whether any assumptions they

made were justified, and whether some other model might fit the data better.

Up to now, what students have been doing could be described as empirical curve

fitting. They have looked for patterns in tables of data and determined a formula to

describe the situation. The tables were not very extensive, and the pattern was not

always perfect. To justify the assumption that exponential functions are the best

models for growth, they need to think about the processes by which living things

grow, and perhaps conclude that it is only reasonable to assume that growth will be

exponential under certain conditions. If growth is constant, however, linear is the best

modeL For bacteria growth to be exponential, temperature and food supply need to be

steady. For populations, the birth and death rates need to be more or less constant.

Students should consider situations as described in the Instructional Activities on the

next page. This will be their first attempt to give a theoretical explanation for the

observed patterns.

Equation> like y = 2', 3', (; J, and (; J, can be dete<mined from table<. Some teal

situations will be more complicated, and finding the appropriate model would be

very difficult to do by hand, so, it is expected that students will use technology.

Students should also consider conducting experiments using calculator- or computer

based laboratories. They should determine the equation using exponential regression

and analyse how well the equation fits the data.

Students should also deal with situations in which the dependent variable decreases in

a gradual way. Students should be encouraged to conduct an experiment of a situation

such as a swinging pendulum to find out how a swing gradually slows down or

measuring the temperature of a cup of coffee to determine how fast it cools. They

could use the CBL or PSL probes to measure the temperature over time, or a motion

detector (or their naked eye) to time or measure the decreasing amplitude of a

pendulum.


Exponential Growth

Worthwhile Tasks for Instruction and/or Assessment Performance (C2/C25/C3/Fl)

1) Suppose that a strain of bacteria has a doubling time of one hour at room temperature if a plentiful food supply is provided. a) If 10 organisms are present at the start, find how many there will be after 0, 1, 2, ...

hours, and enter your results in a table. b) Write an algebraic expression for the number after t hours. c) Doubling times for bacteria vary according to the temperature and the strain of

bacteria. Add more rows to your table to show the growth of bacteria with doubling times of i) 2 hours ii) 3 hours iii) 30 minutes iv) 20 minutes

d) Find an algebraic expression for each of these growth functions. e) Generalize your formula to bacteria with a doubling time of d hours. f) Sketch the graphs of all your functions on the same axes and label each one. Will the

domains and ranges of these functions result in graphs that are continuous or discrete?

g) If Cis the number at the start, and d the doubling time, write down a formula for N, the number of bacteria after t hours.

2) Suppose that, in each unit of time, t, the number of deaths is 5% of the population and the number of births is 15%, so that the increase in each time interval is 10% of the size at the beginning. The time interval may be a year, a month, or a week, depending on the species you are dealing with. a) If there are 1 00 individuals in the population when t = 0, how many will there be

when t= 1, 2, 3, ... ? b) What is the ratio of each of these numbers to the one before it? c) Use this growth ratio to work our a formula for the size of the population at any

timet. d) Go through the same steps to find what the formula would be if the increase per unit of

time was 5%. e) Generalize to the situation where there is an increase of ro/o per unit of time.

3) When Jose and Terri carried out a water cooling experiment, they obtained the results given in the table below.

t 0 2 4 6 8 10 12 14 16 18 20 T 89.0 83.2 78.0 73.7 70.4 67.4 64.8 62.6 60.0 58.0 55 .8

Here tstands for the time in minutes since the experiment began and Tfor the temperature of the water in degrees Celsius. Room temperature was 20°C. a) Graph the data. What do you think will happen to the water temperature if you

wait long enough? b) Add another row to the table showing the difference between the water

temperature and room temperature. How would you test whether an exponential model would fir these numbers?

c) The first three numbers do not appear to fit the same pattern as the rest. Can you suggest a reason for this?

d) Find an exponential function that firs rhe numbers from t = 6 to t = 20 as well as possible.

e) According to this model, what temperature would you expect the water to be after 30 minutes?

4) Kate bought a computer for $2 000, to use in a business she is setting up. If it depreciates at a rate of 30% per year, what will the depreciated value be after one year, two years, .. . five years? Find an expression for its value after n years and show this on a graph . Approximately how long does it take for the value of the computer to reduce to half the initial amount?


EXPONENTIAL GROWTH

Suggested Resources

Barnes, Mary. Investigating

Change: Growth and Decay:

Unit 7. Melbourne:

Curriculum Corporation,

1993.

Brueningsen, Chris, et al.

Real-World Math with the

CBL System. Dallas: Texas

Instruments, 1994.

99

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


wi/1 be expected to


characteristics of exponential





relationships

100


C33/C29 Students should understand that there is a typical shape for an exponential

graph. They should be able to talk about how quickly y-values become very large in

exponential growth situations. If students

wanted to see the results of larger x-values, y y

they would need to reduce the scale on the 10 1.0

y-axis by a huge factor, otherwise the graph 8 0.8

would run right off the page. If graph one 6 0.6

represents y = 2x, then what would be the

value of y if x = 10 or 15. When xis

negative the value for y = 2x soon becomes

so small that it is hard to distinguish the

graph from the x-axis. To get a better idea

-2 2

0.4

0.2

-6 -4 -2

of how the graph behaves for negative values for x , students should extend the x-axis

and reduce the scale on they-axis (see graph 2). Ask students to explain why the two

graphs look so much alike.

Both graphs 1 and 2 are graphs of y = 2x. Students should compare the graphs

y = Y , y = 4X, y = 1 OX withy = 2x and describe what point they all have in common.

When x is positive, which graph lies above the other? When xis negative, which

graph lies above the other? Students should make conjectures from the above

questions and rest them on y = 1.5x andy = 2.SX. Students should compare graphs of

y = 2 - x and y = (; r . They should try to graph y = b X when b is negative. What

happens? Explain.


Exponential Growth


Journaf!PaperandPencif (C33/C29)

1) Explain why all equations of the form y = Ctr where a > 0 and C > 0 and a ;:f:. 1, pass through (0, 1) .

2) Explain why the graph of y = 2.5 x approaches the x-axis more quickly than the graph of y = 1.5 x.

Performance

1 1

3) a) Investigate the graphs of y = 362x ,y = (62 )2x ,y = ( J36r, andy= ( .JGt. Explain

what is happening.

1

b) Based on what you have learned in a) rewrite (52 )Zx three different ways.

1 1

c) Investigate the graph of y = 23x and y = (52 )2x. Explain what is happening.

4) Graph y1

= 2 -x, y2

= 2 x. Harry said that if he graphs y1

= 2 - x for x:::; 0, y2

= Y

for x;::: 0, on the same axes then he gets the graph y3

= x2 + 1. Do you think

Harry is correct? Use mathematical reasoning to correct Harry or defend him.

5) Examine the following tables and indicate which one(s) are suggesting an exponential relationship. Explain your thinking.

X y X y X y

0 0.093 -7 20 000 -6 -3456 0.1875 -6 200 -4 -1504

2 0.375 -5 2 -3 -828 3 0.75 -4 0.02 -2 -352 4 1.5 -3 0.0002 1 -124 5 3 -2 0.000002


EXPONENTIAL GROWTH

Suggested Resources

101

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

C35(Adv) write exponential

functions in transformational

form , and as mapping rules to

visualize and sketch graphs



notation

102

Elaboration -Instructional Strategies/Suggestions C35(Adv)/A7 Students should explore exponential relationships so that when they represent the relation with an equation and/or mapping rule, they will be able to estimate the position of the graph, the orientation, and the behaviour on the coordinate system.

Based on the investigations of the transformations of y = x2 in their previous study, students should explore transformations of the exponential equation y = 2 x.

Students might begin by graphing y = 2 x andy= 2 -x. Have students explore whether they can position one of them onto the other for comparison. Ask them a series of questions that will lead them to discover that y = 2 -x is the image of y = 2 x after a reflection in they-axis, and this transformation can be recorded in a mapping rule: ( x, y) ---7 (-x, y ).

Students should continue to investigate all the transformations as they did for quadratic and trigonometric functions. They should be

. . . 1 ) -2(x+l) able to take an equation hke thts -- (y + 2 = 2 and form a

2

mapping rule (x, Y)-? (-~ x- 1,-2 y- 2!. Most importantly they

should be able to visualize the shape and dcation of the graph on the coordinate system.

Student thinking might go something like this:

• The graph of y = 2 X, y = 3 X, etc. all pass through the point (0, 1).

y -2 -1

This point (0, 1) is called a focal point because the graph looks like it behaves differently on one side than on the other side. For example on the graph y = 2 x to the left of the focal point the graph curves slowly downwards toward the asymptote. On the right of the focal point the graph curves quickly upwards as the xvalues increase. Thus in graphing the equation-~ (y + 2) = 2 - !(x+ '), first notice a negative on both the

2 x and they which indicates that the graph of y = 2 x will be reflected in both they- and x-axis, resulting in the graph curving slowly upward towards the asymptote on the right of the focal point and downward quickly on the left of the focal point.

• The domains include all real numbers( x E R), but the ranges are restricted by the asymptotes ( y < 2jy E R).

To find the image of the focal point (0, 1), students would first consider the reflections. Obviously, only the reflection in thex-axis will have an effect, moving (0, 1) to (0, -1) . Then consider the stretches, and again only the vertical stretch has an effect since the stretch factors multiply the (0, -1) coordinates to locate the stretched image at (0, -2). Then students deal with the translations moving (0, -2) left 1 and down 2 to (-1, -4). Due to the double reflection and the two stretch values, the graph will be increasing, from left to right, rising very sharply (steep slopes for the tangents to the curve) towards the focal point, then continuing to increase at a very slow rate as it approaches the horizontal asymptote, which is two units above the focal point (vertical stretch of2).

To describe the x-values for which the graph(y = Y) increases, students would write

xE R or, in interval notation, x E ( -oo,oo) . Y-values can be described as being

restricted by the asymptotes. Thus, for exponential functions they-values always lie

entirely above or below the horizontal asymptote and can be represented using

inequalities and set notation.


Exponential Growth


Peiformance (C 35(Adv)/A7)

1) An exponential function,.f is defined by f(x) = y = 2 -X+I- 8. a) Write the equation in transformational form. b) Describe the function as a transformation of y = 2x and graph it. c) State its domain and range and the equation of its asymptote. d) Find its zero(es). e) Solve for x if y = 24.

f) Use a calculator to findf(-0 .5) and J(.J3) to two decimal places.

2) An exponential function, g, is defined by g(x) = y = -3 x+I +6. a) Write the equation in transformational form. b) Describe this as a transformation of y = Y and graph it. c) State the domain, the range and the equation of its asymptote. d) Use a calculator to approximate its zero(es) to two decimal places. e) Solve for x if g (x) = -3.

f) Use a calculator to findg (0.5), g( .J2), and g (-2.6).

g) What relation would describe the region above the graph of g? below the graph of g?

3) a) Draw the graph of y = 2x. b) Use your knowledge of transformations to describe in words each of the

following and graph them on the same coordinate system. A:y=2 x+3 B:y+1=2 x C: -y = 2x D : J = 3(2 x-Z) + 5

c) Write the mapping rule for each transformation. d) What transformation will change the asymptote of y = 2x?

4) Find the equation of the image y = 2x under each of the following mappings.

a) (x,y)~(-x,y)

b) (x, y)~(-x,y+l)

c) (x,y) -7 (3x -l,y+ 5) 5) Given that j(x) = y = (1.2) x-

1 - 3.

a) What are the domain and the range of the function? b) For what values of x does [increase? decrease? c) What is the approximate zero off?

d) Describe [as a combination of transformations of y = (1.2) •.

6) For y = 2 x, prove that a horizontal translation of -2 is equivalent to a vertical stretch of 4.

7) What is the equation for the graphat the right? Explain.


EXPONENTIAL GROWTH

Suggested Resources

103

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

C35(Adv) write exponential functions in transformational

form and as mapping rules to

visualize and sketch graphs

C33 analyse and describe the characteristics of exponential and logarithmic relationships

104


C35(Adv)/C33 By exploring equations, tables, and graphs, it should be clear that the exponential graph is either increasing or decreasing and always approaches a horizontal asymptote, e.g., students can see an increasing exponential graph in A and C, but decreasing in B. The horizontal asymptote in A is y = 0, and in B is y = 1, and C is y = 2 (always determined by the vertical translation) . This ensures then that exponential

equations will at the most have one real root, and this will occur only when the graph crosses the x-axis (as in C).

-2 -1 1 2 -1 1 2 3

A B c


Exponential Growth


Performance (C 35(Adv)/C33)

I) Analyse the graph of the function y = -3<x•Zl and answer the following questions:

a) State the domain and range. b) Are there any zeros for the function? If so, what are they? c) Write the equation for the asymptote, if any. d) Does the function increase or decrease with x? Give reasons for your answers.

2) a) Explain why the following equation is not in transformational form j(x) = -3 · 2 (2x -G) + 5

b) Put the above equation in transformational form and describe the transformations of y = 2 x in words and as a mapping rule.

c) State the domain, range, zero(es), and equation for the asymptote. d) Describe the interval for which the graph decreases using symbols.

e) If, for some reason, the domain is restricted to -5:::; x:::; 0 , describe the

corresponding range using set notation.

3) Answer the following questions about the graph drawn on the right. a) Is the curve exponential? How do you know? b) Is it a transfromation of y = 2 x or y = Y? c) Are there any reflections? Describe them. d) Are there any stretches? How do you know? e) State the equation for the curve, then check with your

graphing calculator.

5

-1


EXPONENTIAL GROWTH

Suggested Resources

105

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

B 1 demonstrate an





solving exponential equations

Bl2 apply real number

exponents in expressions and

equations

C24 solve exponential and

logarithmic equations



equations

106


Bl/Bl2/C24/C25 When asked to solve an equation in the form 4 = 2 x students

should understand that the solution is located at the point where y = 4 andy = 2x

intersect, and this can be found using a graph. Students should realize that they are

looking for an x value that when written as an exponent with a base of 2 gives the

result 4 . However, this process may become complicated in equations such as

23x +

5 = 0 .5 . Now it is not as clear what x value would be such that 23x+S would give

1 1 the answer 0 .5. However, students understand that 0 .5 = - and that 2 - 1 = -.

2 2

Therefore, they can rewrite 23x+5 = 0.5 as 23X+5 = 2-1 • Because the bases are equal,

students solve the equation for 3x + 5 = -1 for x .

Many students will understand that a problem occurs when both sides of the equation

cannot be expressed with the same base. Trial and error methods should be explored

leading to the need for logarithms (see next page). A few students would enjoy the

puzzle-solving-like opportunity to unravel equations like:

1) 52x+1 = :JE 4) 9h t = 2T (3xr

2) 4lx-3 -9=55 5) 52x -26(5x)+25=0

3) 92x+ l = 81(27 Y 6) 3x+l + Y = 324

5) and 6) are for advanced students only.

Discussion should focus on multiple representations and justification of each. For

example, ) ~ 2 7 can be exp'e"ed "' ( d' )l m ( d l )' . S 'uden" could "'nhi<

strategy to solve

1) 3 and 2) 3

d 2 =27 d 2 =8

( d± J ~ 3'

I I

(d3 )2 = 642

d 3 = 64 _!_

d 2 =3 d=4

d=9

... continued


Exponential Growth


Pencil and Paper (B 1 /B 12/ C24)

1) Solve the following algebraically:

a) g2x = 24x+l

b) 4x = 2x+7

c) 52x+l =€5

2) Solve for x (Advanced-level students) .

a) 32x -10( 3x) + 9 = 0

b) 3x+l +Y = 324

c) 9h 1 =2T(Yr

I

d) J2 = 81J8 X

3) Find the value for x which ( 4x )( 22x+2 )( 4x+2 )

3Y 512.

4) Solve for the following equations:

a) (irx-3 =(±r+2

b) ~ cl (x-5)3 = B

c) 92x+ l = 81(2r)


EXPONENTIAL GROWTH

Suggested Resources

, .. continued

107

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

Bl demonstrate an





solving exponential equations

Bl2 apply real number

exponents in expressions and

equanons





equations

108


00 0 continued

Discussion with advanced level students might also include recognizing that

2 0 22x = 22x+l 0 For example, when advanced level students are asked to solve

22x+l + 3 0 2x = 5 algebraically they could rewrite as 2 ( 2 lx ) + 3 ° Y - 5 = 0 and again as

2 ( Y Y + 3 · Y - 5 = 0 then factoring:

2. 2x = -5 and 2x = 1

Y =-2 andY =2° 2

x=¢ x =O

Ooo{o}

or by substituting a = 2x; 2a+3a-5 = 0

(2a+5)(a-1)=0

2a+5 = 0

2a=-5

5 a=--2

so, y =-% x=0

and a-l =0 a =l

:o x = 0

{0}

:o{o}

00 0 continued


Exponential Growth


... continued

Pencil and Paper (Bl)

5) What follows are some examples of student work. You are to explain what the student is doing or thinking between step 1 and step 2:

a) b) (x-3)4 =(x+3f Step 1 4x+I =64step 1

[ ' ] 2 2 4x+1 =43 step2

(X- 3 r = (X+ 3) Step 2 :.x+l=3

(x-3f =x+3

x 2 -6x+9=x+3

x 2 -7x+6=0

lOx< 6

PeifOrmance (C25)

6) The table represents population increases from 1955 to 1990. The numbers are in billions.

Year 1955 1960 1965 1970 1975 1980 1985 1990 Population 2.7 2.98 3.29 2.67 4.01 4.43 4.84 5.29

a) Graph the relationship using the values in the above table and determine the equation that best represents the relationship.

b) Using the equation, determine the population in the year 2010. c) Explain how you would use the equation to predict how long it would take for

the population in year 2010 to double, assuming it continues to grow at this rate.


EXPONENTIAL GROWTH

Suggested Resources

109

EXPON ENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

Cl9 demonstrate an

understanding, algebraically

and graphically, that the

inverse of an exponential

function is a logarithmic

function








notations

110


Cl9 As students explore and solve exponential equations in various forms, eventually

they will need to solve for the exponent where they are unable to find common base of

all terms. When this has happened earlier in their learning, students simply read a

value from the graph or used trial and error methods.

It is now time for students to explore this concept in more depth. Ask students to

find x when Y = 9. Students will quickly say that x = 2 because J2 = 9. Now ask

students to find x when 3x = 7 using a trial and error procedure. (The graphing

calculator is a tool students could use here by drawing the graphs y = 3x andy= 7 and

examining the interaction points.

Cl9/CIIIA7 To symbolize what the calculator has done, students need to be

introduced to the term 'logarithm.' This name is used to describe the inverse function

of an exponential function with the same base.

• Ask students to plot y = 10 x from a table where -5 ~ x ~ 1 and state the domain

and range. • Have them then reflect y = 10 x across y = x and plot the resulting curve. Have

them state the domain and range again and compare. • Discuss with students why they should draw the inverse y = 10 x by reversing the

coordinates in the table for y = 10 x and plotting them (y = x is the line of symmetry between the graphs of a function and its inverse) .

• Use the 1 ox feature (2nd, Log on the Tl-83) , to enter 1 0/\(2. 1) and hit enter. Then have them press Log and ANS [answer: (-2nd, (-))] and hit Enter.

• Ask them to describe what has happened discuss how one (1 ox or LOG) is a function that undoes what the other function did (the term for this is "inverse function") explain how they can show the same process on the graphs they drew use the 'log' button on their calculator and check their table and graph by finding the logarithm of each of the x-values in the table create the same two graphs using technology, then examine the table in the TABLE feature

Students should develop the equation for this new graph as y = log x (log x and log10

x

mean the same-the base 10 is assumed). They start with y= 1 OX, switch the x andy

to get x = 1 QY, then solve for y by using the symbol for logarithm (log) . Since y was

the exponent, and log means exponent, then y = log x.

Similarly if they were to begin withy= 2", the inverse function would be written

y = log2

x. Students need to understand how to evaluate logJ using their calculator

since the 'log' button assumes base 10, (see Bl3, next page).

ATLANTIC CANADA MATHEMATICS CURRICU LUM: MATHEMATICS 3204/3205

Exponential Growth


Performance (CI 9/CIIIA7)

1) a) Find the equation of the inverse function, h, defined by h(x) = 7 2x and graph it.

b) Is this inverse a function? c) Find the zero(es) of h -I. d) Could you graph h - I by graphing h(x) first? How?

2) a) Sketch the graph of J(x) = 1 OX for these values of x. x = 0.1, 0.2, 0 .3, 0.4, 0.5 , 0.6, 0 .7, 0.8, 0.9, 1.0.

b) Estimate from the graph the values of x for y = 1, 2, 3, ... , 10. c) Construct a table of values using the values of yin (b) for x and the values of x

in (b) for y . Plot the ordered pairs obtained and join them with a smooth curve. d) What relation would you use to describe the graph expressing y in terms of x? e) State the domain and the range of this graph.

Pencil and Paper (CI9/CIIIA7)

For questions 3 and 4 below, use the graph of y = log 10

x shown. Give your answers correct to two decimal places and check with a calculator.

y

14

1.2

10

0.8

06

0.4

02

2 3 4 5 6 7 B 9 10 11 12 13 14 15 16 -Q.2

-Q4

3) Find an approximate value for each of the following. a) log

10 4 b) log

10 7

c) log 10

8.5 d) log 10

3.5

4) What is the approximate value of x for each of the following? a) log

10 x = 0 b) log

10 x = 1 c) log

10 x = 0.7

d) log 10

x = 0.4 e) log 10

x = 0.65 f) log 10

x = 0.35

5) Explain that the equation of the image of y = log 5

x under a vertical stretch of 2 followed by a horizontal translation of -1 and then a vertical translation of 2 is y = 2log

5 (x + 1) + 2.


EXPONENTIAL GROWTH

Suggested Resources

111

EXPONENTIAL GROWTH

Exponential Growth

Outcomes


will be expected to

B1 3 demonstrate an understanding of the

properties oflogarithms and

apply them



Elaboration -Instructional Strategies/Suggestions B13 Before calculators and computers were readily available, logarithms of numbers

were an indispensable aid to scientific calculations. Although this role has been taken

over by machines, logarithmic functions are still crucial models for many important

scientific phenomena. It is still important to be comfortable with the algebraic

properties oflogarithms as well as exponential functions. Students should strive for mental math capabilities with the use of many of these properties. Time should be

devoted to helping students develop strategies. For example, students should be able

to calculate mentally log}2 using the strategy of answering "2 to what power equals 32?".

Properties to be developed (assume x andy> 0)

1) If y = tr, then x = logay 2) log.2 + log) = log.6 then generalize to log x + logy= log xy and describe this in

words.

X 3) logx -logy= log

y

4) log 9 = 2 log 3 and generalize this to log x2 = 2log x

I logx

5) og x=--a loga

I 6) log-= -logp

p

7) log. a =l 8) logh lr = x

I 9) logba=--

log. b

These properties are useful when simplifYing logarithmic expressions and trying to solve equations requiring logarithms. Since solving logarithmic eq uations algebraically

requires students to use the strategy of simplifYing the equation to a single 'log' term of the same base on both sides, students would use various properties above to

simplifY the expression.

Students should also be able to solve problems of the type:

a) log 6 (x+I) + log6 x=I

b) I 03x-s = 20

112 ATLANTIC CANADA MATHEMATICS CURRICULUM . MATHEMATICS 3204/3205

Exponential Growth


Pencil and Paper {Bl3/C24)

I) Express each in logarithmic form.

b) _91 = 3-2

a) 9 = 3 2

2) Express each in exponential form.

a) 7 = log 2

128 b) -6 = log2 ( ~) 3) Evaluate each expression.

a) log 9

81

e) log_7

i) log" a12

b) log6

6

f) log 81

9

j) logsO

c) log 3

4 = x

c) log2

1

g) log 2

(2 3)

4) If log 10

4 = 0.602, find log 10

16 without using a calculator.

5) Iflog bx = p and log by = q, find Iogb (xy), x andy> 0.

6) If log. b = xand log . c = y, find Jog, ( b: } 1

Iogb x 7) Prove that og x = --

a Iogb a

d) 27~ =3

d) ± = log2 ( ifi)

d) log3 ( ~) h) log . ax

8) Express each of the following as a single logarithm in simplest form . a) Slog

4 2 b) log s 48 -log s 12 +log s 4

c) 3 log 6 15 -log 6 25 d) log ax + log a y-log az

1 2 e) log b 2x + 3(log bx -log by) f) llog" x- 3loga Y

9) Evaluate.

a) log 3 9 + log 3 27 + log 4 64 + log4 ( 1

1

6) + log3 (;) + log 10 ( 1~)

b) log 2 32 -log{ 3

1

2) + log4 8 -log8 16

c) 3 log 2 4+2log3 9+log 10 (0.1) - log{;)

10) Express each logarithm in terms of m and n where log b x = m and log 6 y = n.

11) Solve for x:

a) log.J2 8 = x

b) log b {xy2) c) logb(R )

b) 1

x = log 2 8 - log - + 2logsl25 9


d) log, J;

EXPONENTIAL GROWTH

Suggested Resources

113

EXPONENTIAL GROWTH

114 ATLANTIC CANADA MATHEMATICS CURRICULUM: MATHEMATICS 3204/3205

Unit 4 Circle Geometry

(30 °/o)

CIRLCE GEOMETRY

Circle Geometry

Outcomes


wiLl be expected to

E4 apply properties of circles

E5 apply inductive reasoning

to make conjectures in

geomerric situations

Ell write proofs using

various axiomatic systems and

assess the validity of

deductive argumenrs

E7 investigate and make and

prove conjectures associated

with chord properties of

circles

EB investigate and make and


with angle relationships in

circles



with tangent properties of

circles

El2 demonsrrate an


of converse

116


E4 Geometry is a rich field of mathematical study. The world around us is inherently

geometric, and hummankind's creations most often reflect geometric principles. The

concrete and visual nature of geometry resonates with certain learning styles, and

geometry's pervasiveness in our environment facilitates connecting the study of

geometry to meaningful, real-world situations. This is as true for circle geometry, the

focus of this unit, as for geometry in general. Whether determining the correct

location for handles on a bucket, finding the centre of a circle in an irrigation project,

or determining the length of a tangent to the earth from an orbiting satellite,

properties of circles (and lines, line segments, and/or angles associated with them)

come into play.

Elll£5 Geometric figures such as segments, lines, angles, polygons, circles, and

planes are each sets of points that are subsets of the universal set called space. In

synthetic (Euclidean) geometry, these geometric figures can be drawn anywhere on a

plane in space; in analytical (coordinate) geometry, a reference system is added, and

important points on the figures are assigned coordinates. Using transformations,

these figures-with or without coordinates-can be moved in space by following

specific rules. In all perspectives, students seek to discover patterns among figures or

within a fixed figure.

Students need many opportunities to explore geometric situations, look for common

elements (or patterns) in them, and make appropriate conjectures. They also need to

reach an understanding that, while this inductive process of observing multiple cases

and conjecturing seems to imply the truth of a relationship, deductive reasoning is

required 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 a variety of modes of proof, with the understanding

that a logical argument can take many different forms. This implies that students

should experience Euclidean, coordinate, and transformational approaches and

develop an appreciation that each can be advantageous in certain situations. This unit

provides the opportunity for students to develop some proficiency with all three

modes of proof If time should be a factor, however, it is sufficient for students to

focus on analytical arguments and either Euclidean or transformational.

E7 /E8/E9/E5/Ell/E12 In particular, contexts will be explored, and theorems

conjectured, proven and applied, with respect to chord properties in circles, inscribed

and central angle relationships, and tangents to circles. The treatment of these circle

topics is not intended to be exhaustive, but is determined to a significant extent by

the contexts examined. It should also be noted that the concept of the converse of a

theorem will also be explored in relation to some of the theorems developed.


Circle Geometry


Peiformance (E4/E5/E7)

1) Activity: a) Begin with a circle of any size (given to the student) .

Fold the circle in half-make a crease. Open up the circle and fold in half differently.

Open up the circle and investigate the intersection point-compare with your classmates-make a conjecture.

b) Begin with a circle of any size (given to the student) . Make a fold anywhere on the circle, make a crease. Repeat the first step with a second fold and crease. Mark the first fold AB at its end point, the second CD. Fold A onto B, make a crease. Open the circle, fold C onto D, make a crease. Investigate the intersection of the last two creases-compare with your classmates-make a conjecture.

c) 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-compare with your classmates-make a conjecture.

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

Make fivefolds creating 5 chords all of equal length (fold into the centre).

Investigate the distance each is from the centre-compare with your classmates-make a conjecture.

Peiformance (E4/E5/E9)

e) Begin with a circle on a rectangular sheet of paper and mark the centre. Make a fold at 3 different points on the circumference to produce creases that touch the circle at only those three points. Join the points to the centre and investigate the angles formed between the radii and the tangents-make a conjecture. Place a coordinate system over this situation, or transfer it to a coordinate system, find the slopes of the radius and the tangent to it, compare with your classmates, and make a conjecture.

Peiformance (E4/Ell/E8)

2) IfP is the centre of semicircle, PR bisects AC and PQ bisects Be , prove PR is perpendicular to PQ.

Peiformance (E12/Ell)

3) State and prove the converse of

A~ A P 8

If a point on the hypotenuse of a right triangle is equidistant to all three vertices, then it is the midpoint of the hypotenuse.


CIRCLE GEOMETRY

Suggested Resources

117

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to



geometric situations




circles

El2 demonstrate an


of converse




deductive arguments


118


E5/E7 Students need to begin their study of circles by exploring patterns and making

and verifYing conjectures. They might begin their exploration with an activity like

the following:

0 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 diameter and explain how you located the centre (C). What type of triangle must PQC be? Explain.

• Construct three more different circles that pass through P and Q. Name their centres D, E, and F.

Fold Ponto Q, making a crease to indicate the fold line. What do you notice about the points C, D, E, and P. Name the point where the crease intersects P, Q, as M. IsM the midpoint of PQ? JustifY your answer.

Is PQ l. to the fold line? How do you know?

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

E5/E7/El2 While exploring patterns (as in the previous activity), students might use

paper-folding techniques and/or measurement tools like rulers, dividers, compasses,

and protractors. In so doing, they will be using both transformational and Euclidean

techniques. They will also be using inductive reasoning to make conjectures such as

• any point that is equidistant from two points on a circle must be on the perpendicular 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 be equidistant from the end points of that chord.

Ell/E4 Moving from conjecturing to proving conjectures in general (e.g., proving

theorems), and applying new theorems to calculate or prove other geometric facts ,

may be a large step for many students. Teachers will need to model the thinking

processes necessary to generate proofs. As well, it may be necessary to spend time

reaquainting students with the geometric properties and theorems with which they

are already familiar (e.g., congruent triangles, angle sum of a triangle, vertically

opposite angles, parallel line theorems). Seep. 122 for a further elaboration.


Circle Geometry


Performance (E5/E4/E7 /Ell)

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

thinking. b) Prove your conjecture in (a).

2) Construct a large circle and two non-parallel congruent chords that are not diameters. a) Compare their distance to the centre of the circle. b) Write your findings in (a) as a conjecture. c) Test your conjecture on other circles. d) Prove your conjecture.

Performance (E12)

3) a) Restate the conjecture you made in question 2 above in an "if ... then .. . " form . b) State the converse of this conjecture. c) Is the converse true? Explain.

4) State a theorem related to geometry whose inverse is not true.

Performance (E4)

5) Use a circular object to trace a circle onto your paper. Without using a compass, locate the centre of the circle.

Performance (E4/E7)

6) A piece of circular plate was recently dug up on an island in the Mediterranean. The discoverer of the plate wishes to calculate the diameter of the original plate. Describe how he could do this.


CIRCLE GEOMETRY

Suggested Resources

Hirsch, Christian R. ed.

Curriculum and Evaluation

Standards for School

Matheamtics. Addenda

Series. A Cone Curriculum.

Reston, VA: NCTM, 1992.

119

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to

E12 demonstrate an


of converse




circles




circles

120


E12/E7 Once students begin to articulate conjectures and try to establish them as

theorems, it would be appropriate to introduce them to the concept of converse.

Essentially, a converse is an opposite, or something reversed in order or action.

Consequently, the converse of a conjecture or theorem will be reversed in order and/

or action with respect to the original.

Students may find, when they examine the conjectures that they made in their initial

explorations, that some are indeed converses of others. They may find this easiest to

do if they adopt the conventional, conditional form for statements of conjectures/

theorems, e.g. , the "if ... , then .. . " statement. For example, in this format the

statements from p. 116 become

• If a point is equidistant from two points on a circle, then it lies on the perpendicular bisector of the chord joining those two points If a point lies on the perpendicular bisector of a chord of a circle, then it is equidistant from the end points of that chord.

E12/E8/E7 In general, students should understand that the converse of any "if p, then q,"statement is "if q, then p." As well, it is critical that students understand that

the truth of any theorem does not necessarily imply the truth of its converse. In some

cases a converse is also true, in other cases it is not, so students should always test the

truth of a converse. For example, they might write the converse of "If an angle is

inscribed in a semi-circle, then it is a right angle" and see that the converse is not true

in general. The discussions arising out of the examination of these types of examples

encourage logical thinking.

Students should know that if a statement and its converse are true, it can be stated as

an "if and only if"(iff) situation. This gives rise to sufficient conditions: e.g. , points

lie on the perpendicular bisector of a chord iff they are equidistant from the end

points of the chord. Students should realize that it is sufficient to prove a line is a

perpendicular bisector of a chord by proving two points on it are equidistant from

the endpoints of the chord.

Converse is not a concept that requires extensive attention in its own right. It should

be addressed from time to time as it comes up throughout the unit, with a view to

students' developing a clear understanding of the concept.


Circle Geometry

Worthwhile Tasks for Instruction and/or Assessment Performance (E12/E7)

1) State the converse of each of the following. If you think the converse is true, rewrite using 'if and only if" (iff). If you think it is false, explain why. a) If a triangle has rwo angles of equal measure, then it is isosceles. b) If rwo triangles are congruent, then their corresponding angles are congruent. c) A quadrilateral with four axes of symmetry is a square. d) Every square has four sides of equal length. e) The centre of a circle lies on the perpendicular bisector of a chord of that

circle. f) A tangent line is perpendicular to the radius of a circle.

2) State and prove the converse of: If a point on the hypotenuse of a right triangle is equidistant to all three vertices, then it is the midpoint of the hypotenuse.

3) Prove the converse of this statement: If a line passes through the midpoint of a chord on a circle, then it is equidistant from the endpoints of the chord

4) Make a statement and state its converse so that the original statement is true but

the converse is not. State a theorem in geometry that you know is true but whose converse Is not.


CIRCLE GEOMETRY

Suggested Resources

121

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to





deductive arguments

122


E4/Ell .A5 previously indicated, it may be necessary to refresh students' knowledge of

geometric properties and theorems (e.g., congruent triangles) while building strength

with respect to writing deductive arguments. This may be accomplished by applying

deductive reasoning in situations in which students are asked to apply their knowledge

of geometry to i) determine specific angle measures and/ or ii) prove geometric facts or

theorems in general. This is an opportunity to expose students to both Euclidean and

transformational forms of proof.

Sometimes students might be given some angle measures and be asked to find and

verify that another angle has a certain measure. For example, given the situation

below, find and verify that the angle CDE must be 78°. Students could begin with the

given 51 o angle and using vertically opposite angles and equal angles in an isosceles

triangle, then get mL CED = 78° .

In a slightly more general situation, students could be asked

to prove that AC = DC Using 180° rotation, centre C,

students could state that B HE because B- C- E and

BC = CE then A~ ED because L.ABC = L.CED, so

A ~ D because AB = DE since Cis the turn centre

AC ~CD and :. AC =CD because in a rotation the image must equal the object.

A

~ B c"""JE

D

When students are presented with arguments that are not valid they should be able to

identify the flaw. For example:

Given: AD and BC intersect at E.

Jon argues that AB = CD. His argument goes like this:

L.l = L.2 because vertically opposite angles are equal,

AE =ED, CE =ED because of equal radii.

So the triangles are congruent and AB = CD since they are

corresponding parts of congruent triangles. Find the flaw.


Circle Geometry


Peiformance (£4/Ell)

1) Prove that the incentre of a triangle is the centre of the inscribed circle of that triangle.

2) Prove that the circumcentre of a triangle is the centre of the circle circumscribed on that triangle.

3) Prove that the centre of a circle is the intersection of the perpendicular bisectors of two chords.

4) Prove that two chords equidistant to the centre have the same length. 5) Prove that a tangent to a circle is perpendicular to the radius at the point of

tangency. 6) Use congruent triangles to prove that tangent segments to a circle from a point

outside the circle are equal in length.

7) If the sides of a quadrilateral ABCD are tangent to the circle, show that AB + DC = AD + BC.

8) Find the flaw(s) in the following proof

Prove: L B = L C

Proof: for a rotation, centre E

A~ D(AE = ED )

B~ C(ABI!ED)

: .LB=LC

journal

9) D escribe how the following idea could be proved:

B

~ A E~D

c

The centre of any circle is the intersection of the perpendicular bisectors of any two no n-parallel chords in the circle.

ATLANTIC CANADA MATH EMATICS CU RR ICULUM: M ATHEMATICS 3204/3205

CIRCLE GEOM ETRY

Suggested Resources

123

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to



assess the validity of deductive

arguments




circles

124


Ell/E7 Once students have been reaquainted with the necessary geometric ideas,

they can move on to applying them in the context of properties of circles, beginning

with chord properties.

Students should write proofs using deductive arguments and explain the validity of

the arguments. For example, in proving the conjecture

• Any point that is equidistant from two points on a circle must be on the perpendicular bisector of the chord joining those two pomts

• students might say .. . "Since AP = AQ (A is equidistant to two points) , then the triangle APQ is isosceles and has one line of

symmetry passing through the vertex angle, and LA is the vertex

angle.

~ Q

"A property of symmetry says that this line of symmetry must be the perpendicular

bisector of the base of an isosceles triangle, thus proving the conjecture."

Alternatively, students might use a Euclidean version of the same proof:

"Join A to M, where M is the midpoint of segment PQ; the triangles would now be

congruent since there are three pairs of corresponding sides congruent (SSS). Since

the triangles are congruent, then the corresponding angles at M inside the triangles are

congruent, and they are supplementary, so each is a right angle (two 90s make 180).

Since the angles at Mare right angles, then AM is perpendicular to PQ at its

midpoint, and the conjecture is proved."

By way of a second example, consider the following:

Given that P and Q are points on a circle with centre C, students could fold so that

m\ 0

p \ c

\

\ \

\

CM .lPQ .

P ---7 Q . Since PC= CQ (Cis the centre of the circle), C folds onto

itself,

C ---7 C, e.g., C must be on the mirror line. Properties of reflection say

that the mirror line must be perpendicular to a line joining a point to

its image (P and Q). So if the intersection point is named M , then

Some students might want to prove ~:>.CMP =~:>.CMQ in order to prove CM .l PQ .

Their proof may look like this:

Let M be the midpoint of PQ

~:>.CMP ~CMQ (555)

LCMP==LCMQ (~:>.'s :=)

. ·. mL CMP = 90° = mLMCQ (congruent and supplementary)

:. CM .l PQ, and CM bisects PQ (M is the midpoint)


Circle Geometry


Peiformance (El l/E7)

I) Mary said to Beth: "I can draw a segment inside the circle that is equal in length to one half the diameter without measuring it." Beth was impressed and insisted that Mary show her work. Mary told Beth to draw any circle with diameter AB then to draw two more chords to make the triangle ABC. She told Beth that, if she connects the two midpoints of the two chords just drawn, the segment joining them would be one half the length of the diameter. Explain how you know that the segment joining the midpoints would be one-half the diameter.

2) Lou, who was listening to and watching Mary and Beth, was startled because, he exclaimed, he now knew how to prove that the segments joining the midpoints of any quadrilateral form a parallelogram. How can Lou do this?

3) Draw a circle. Draw two chords of the circle of unequal length. Which is closer to the centre of the circle? Prove it.

4) In designing various logos that use circles, Frank wanted to make sure that what seemed to look correct really was correct. Help him prove the following relationships:

a) Given: OBIIPD

0 and Pare centres of circles

Prove: LOAB = LPCD

b) Given: OAJJPD

0 and Pare centres of circles

Prove: LOAB = LPCD

c) Given: ABJJCD 0 is the centre of both circles

Prove: LOAB = LC

5) 0 and Pare the centres of the circles. a) Why is the figure OBPA a kite? b) Explain why AB ..LOP

c) What other conclusions are valid?

6) Given that 0 is the centre of the circle, and the two central angles are congruent, prove AB=MN

~ ~

ATLANTIC CANADA MATHEMATI CS CURRICU LUM: MATHEMATICS 3204/3205

CIRCLE GEOMETRY

Suggested Resources

125

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to

D 1 develop and apply

formulas for distance and

midpoint



geometric situations




arguments

126


Dl It is common for students to use measurement to attempt to prove conjectures.

However, for generalization, they need to develop a way of generating length or

distance algebraically. They might begin by contrasting the ease of determining the

lengths of horizontal and vertical lines (on a coordinate system) with the difficulty of

finding the lengths of oblique lines. (For example, they might compare finding the

distance of the point (3, 0) from the origin as compared to finding the distance of

(3, 5) from the origin.) The connection between the Pythagorean Theorem and the

distance formula needs to be emphasized and understood, so that students can

connect this new formula to previous knowledge. In the same way, the calculation of

the midpoint can be connected to averaging.

. ) 2 2 . . ( x , + x2 y, + Yz) Distance = (x2 - x,) + (Yz - J 1) M1dpomt = --2

- ,--2

-

E5/Ell/D 1 In circles, there are several opportunities for line segments to be

perpendicular. Opportunity should be provided for students to conjecture that, when

the slopes of two segments are negative reciprocals, the two segments are

perpendicular. Sometimes in coordinate proof, students can benefit from knowing

that the product of the slope of an oblique line and its reciprocal is -1. This may

sometimes be easier for students to see than the fact that they are negative reciprocals

of each other. Once students have made such a conjecture, it needs to be proven.

Students could then apply parallel properties to prove that, given four points, certain

segments are parallel and others are perpendicular, while others are neither. Students

can also prove that some quadrilaterals are parallelograms and some are rectangles or

rhombus.

The coordinate system can also be used to help generalize properties. For example, in

proving that the diagonals of a rectangle are congruent, students would assign variable

coordinates to the four vertices, then express the length of the diagonals DB and AC

D (O,b) c

A (0,0) 8

(a,b)

(a,O)

as

DB=~(a- 0) 2 + (0- b)2

= .Ja2 + bz

AC=~(a- 0)2 + (b-Ol =.Ja2 + b2

and conclude that these lengths must be equal.


Circle Geometry

Worthwhile Tasks for Instruction and/ or Assessment

Performance (D 1/Ell)

1) A sprinkler head is positioned on the infield so that the spray of the water soaks the entire field that lies within the 1 0-metre radius of the head. Assume that a Cartesian coordinate system has been placed over the field and the sprinkler head is at the coordinateS (3, -1). Determine whether or not the new trees at the following locations will get wet or not? a) A (11, 4) b) B (-3, -9) c) C (12, -6) d) D (--4, -8)

2) Another sprinkler head is positioned at P (-2, 6) . A tree is on the circumference at (5.5, 7). What is the area of the ground that gets wet?

3) These trees P (3, 6), Q (6, -1) , and R (-1 , -4) lie on the circumference of the water circle. Prove that C (1 , 1) is the location of the sprinkler head.

4) Determine the coordinates on the field so that the planting of trees A , B, and C lies on the same line and between trees M (-1 7, -8) and N (47, 20) so that segment MN is divided into four equal parts.

Performance (D IIE5/EII)

5) Use Geo-strips to construct a parallelogram. Investigate the many parallelograms and the lengths of their diagonals that can be formed by repositioning the Geostn ps. a) Prove that the diagonals of a parallelogram are not congruent. b) Prove that the diagonals of a rhombus are congruent.

6) Given C1

: 2y- 6x + 2 = 0 C2

: 5x + 5y - 15 = 0 the equations for two chords of a circle, prove that the intersection of these chords is the centre of the circle that contains a diameter that runs from A (--4, 2) to B (6, 2).

7) Given that the radius bisects the central angle in a circle, prove that it bisects the arc subtending the central angle.

8) Given the diagram, ask students to make up a question to find the coordinates for the point P.

journal (DI /E5/EII)

9) Describe how the procedure for calculating the distance between two points, given their coordinates, is similar to the procedure for calculating the slope of the line that joins the two points. Describe how it is different.

ATLANTIC CANADA MATHEMATICS CURRICU LUM. MATHEMATICS 3204/3205

CIRCLE GEOMETRY

Suggested Resources

127

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to




arguments

D 1 develop and apply

formulas for distance and

midpoint

128


Ell/Dl Students might revisit statements that they have already provided using

Euclidean or transformational proofs. For example, students have already shown that

points equidistant to the two points P and Q (see £5, p. 126) are on a line .l to PQ and passing through the midpoint. Using coordinate geometry to assign coordinates

to P and Q ask students to find the coordinates for the centre points C, D, E, and F,

knowing that radii of a circle are equal. Find the slope of PQ and FC. This should

lead to a conjecture that there is a perpendicular relationship of PQ and CM through

midpointM.

PF = .j(x- 0) 2 + (2- 0)2

= ..Jx2 + 4

QF = .J(x-8)2 +(2-0f

= ..J x 2 - I6x + 64 + 4

= ..J x 2 - I6x + 68

Assume F is the centre of the circle:

I I

C (x, 6)

D (x, S) I

/ ll E(x,3)

/ " / --T -.E. (x';'s~ / _....._..... I -- '-.. Pe_..... 1 --c...g

(0, 0) I M (8, 0)

I

..Jx2 +4 =..Jx2 -l6x+68

.·.x =4, and F's coordinates are (4,2)

rise 0 slope ofPQ=- =- = 0

run 8 4

slope ofFC=- = not defined 0

:.FC l. PQ

Students will re-examine these statements and attempt to prove them using

coordinate arguments.

ATLANTIC CANADA MATHEMATICS CURRICULUM: MATHEM ATICS 3204/3205

Circle Geometry

Worthwhile Tasks for Instruction and/or Assessment Performance (Ell/D l)

1) A triangle has vertices R (17, 16), 5 (1, 4), and T (7, -4). a) Prove that the triangle is right-angled. b) M is the midpoint of RT Prove that a circle with centre M and passing

through R, also passes through T and 5.

2) Find the equation for the set of all points that are equidistant from ( 6, 2) and (I, -5).

3) Given a right triangle with vertices A (1, 4) and B (9, 3), the third vertex Cis on the x-axis. If side AB is the hypotenuse, find the coordinates of C. Is there more than one answer? Explain.

4) Ralph decided to use coordinate geometry to help him solve his problem. He located the middle of a beam at M (2, 3) . He was able to locate approximate positions for the endpoints P and Q. He knew the x-value for the point P (-4, y) , and they-value for the point Q (x, -2). Find the values for x andy.

5) A ceiling support beam is constructed from several congruent isosceles triangles. When the midpoints of one of the triangles are joined, will the new triangle also be isosceles?

6) Create a real-life problem using the diagram on the right. 1-1.8) A 13.8)

(-3,5)~(5, 5)

7) Using transformations, prove that the midpoint, M, of a line segment PQ with

endpoints P (a, b), and Q (c, d) is M( a~ c' b: d). 8) Two designers are using a Cartesian plane to design a large plus sign to hang in the

operations room. The plus sign is made using two perpendicular lines whose equations are almost completely determined: x- 2y = -8, and kx- y = -3 (x + 1). Determine the value for k. Is there more than one value for k? Explain.

9) Determine the ratio of the sum of the lengths of the three altitudes to the perimeter of the triangle whose vertices are P (0, 0), Q (4, 3), and R (-2, 5). Do you think that this ratio is the same for any triangle? Investigate.


CIRCLE GEOMETRY

Suggested Resources

129

CIRLCE GEOMETRY

Circle Geometry

Outcomes


wiLL be expected to




arguments

EB investigate and make and



circles



with tangent properties of

circles

130


Ell/E8/E9 Students will continue to apply their knowledge of axioms and algebra to

construct logical arguments with respect to other properties of circles, in particular

angles in circles and tangents to circles. For example, to prove that the inscribed angle

is half the central angle sub tended by the same arc, students might use Euclidean

geometry:

Draw the line through BO. In triangleAOB: OB = OA because

of equal radii mLA = mLOBA ="a" because triangle AOB is

isosceles. Thus, mLAOD = 2a since the exterior angle of a

triangle is the sum of the two remote interior angles.

mLDOC = 2b, and

mLAOC = 2a + 2b

=2(a +b) Similarly, mLABC =a + b

: . mLAOC = 2mLABC

B

Students could be asked to make the logical argument that an inscribed angle in a

semicircle is 90°. Some students might say that an inscribed angle is half of the central

angle sub tended by the same arc and since the central angle is 180°, the inscribed angle

must be 90°. Other students might construct two isosceles triangles as in the above

diagram and show that when 2a + 2b = 180° results the equation a+ b = 90° is logical

for LABC. The "proofs" need not be long and involved. What is important is that

they are based on agreed-upon axioms within the geometry system (Euclidean,

transformational, coordinate) being used.


Circle Geometry


Performance (Ell/E8/E9)

1) Find the diameter for the circle with centre Q.

A

2) Given: LR is a right angle T, U, and V are points

of tangency with the circle centre P.

1 Prove: r = 2( QR + RS- QS)

when r is the radius of circle P.

3) Prove: a) The central angle of a circle is twice the measure of the inscribed angle on the

same arc.

b) The angle inscribed in a semi-circle is a right angle.

4) Two circles with centres P and Q are tangent at S. Prove that P, S, and Q are collinear points.

5) In the diagram, each circle is tangent to the other two. If AB = 10 em, AC = 14 em, and BC = 18 em, find the radius of each circle.

6) AB is a semicircle with centre C, PQ is concentric

with AB, EC ..lAB, and CD ..l CF. Prove

mAD+ mQT = mEF +mRS . DtM A P C Q 8

7) Construct two non-parallel, non-congruent chords on a circle of any radius. Connect the endpoints of the chords with segments so that the segments intersect. Measure the four angles formed at the circumference. a) Make a conjecture about two angles subtended by the same arc. b) Construct a central angle and compare the measures of a central angle with the

inscribed angle subtended by the same arc. Make a conjecture and check it with other central angles in the diagram.

c) Prove the conjecture in (b), then use that proof to prove the conjecture in (a).


CIRCLE GEOMETRY

Suggested Resources

McKillop, David W et al. Pre- Calculus Mathematics

One. Toronto: Nelson, 1992

131

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to


El5 solve problems involving

the equations and

characteristics of circles and

ellipses

132


E4/E15 Many problems can be solved with the drawing of a circle or circles. Students

will explore how technology (especially using the Draw Mode on graphing calculators)

can be used to construct circles.

All students should be able to attempt to solve problems involving circles such as the

following:

The planners of the arena want to have a smaller rectangular grassed area on which

games like croquet, badminton, tennis, dodge ball, etc., might be played. Suppose the

field might have dimensions 75 m by 50 m. The groundskeeper for the field wants to position three sprinklers. Each sprinkler throws water out over a semi-circular path.

He wants to position the three water heads on three of the four boundaries of the

field so that every place in the field will be exposed to water. He places one at the

centre of the front 50 m wall, and one on each of the 75-m walls, 55 m back from the front wall. Will sprinkler heads that cover semi-circles up to 30 m be adequate, or will

he have to purchase the expensive 35-m size or even more expensive 40-m size?

Students might begin to explore this problem by drawing circles (or by using their

graphing calculators) . If using calculators, they would set their window to represent the field dimensions, then draw circles using the three sprinkler head locations as centres. In the Draw menu, select 9: circle (0, 25, 30), then push Enter. This will draw a circle centre (0, 25) radius 30 m.


Circle Geometry


Performance (E4/E15)

1) How many circular pipes, each with an inside diameter of 10 em, will carry the same amount of water as a pipe with an inside diameter of 60 em?

2) By how much does the radius of a circle increase when the circumference is increased from 20 em to 25 em?

3) A 16-cm chord is 15 em from the centre of a circle. What is the radius of this circle?

4) An 18-cm chord is perpendicular to the radius of a circle. The distance from the intersection of the chord and the radius to the outer end of the radius is 3 em. What is the length of the radius?

5) Two circles are internally tangent at A . Cis the centre of the larger circle. BA is perpendicular to CF, EF = 5 em and BD = 9 em. What is the length of the diameter of the smaller circle?

6) Two gears have radii 5 em and 3 em. How many times must the smaller gear be turned in order for the arrows on each gear to align again?

F

G

7) For the past several years on opening day of lobster season, the weather and other circumstances have caused life-threatening incidents. This year the air sea rescue helicopters are to be placed so that every point in the 90-km square region can be reached within 20 minutes. In 20 minutes these helicopters can travel up to 36 km. Three plans are proposed. Assuming that you will use an appropriate window on your calculator: PlanA

Five helicopters

One in the middle

One at each corner

Plan B

Four helicopters at (20, 20), (20, 70), (70, 20), (70, 70)

PlanC

Four helicopters at (25, 25), (25, 65) , (65, 25), (65, 65)

How can you use the circle-drawing feature of your graphing calculator to evaluate these three plans? Do it. Which is best? Which is worst? Why? Can you create a better plan?


CIRCLE GEOMETRY

Suggested Resources

McKillop, David W et al.

Pre- Calculus Mathematics

One. Toronto: Nelson, 1992

133

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to

E 15 solve problems involving

the equations and characteristics of circles and

ellipses


D I develop and apply formulas for distance and

midpoint

134


EI5/E4/DI In exploring the sprinkler system described in the elaboration for E15, p.

130, it would be helpful to have other ways to decide whether or not given points are

within the circular boundary of the spray given off by the sprinkler heads. For example, have students determine if the following points are within the 10 unit

boundary, on the boundary, or outside the boundary of the circular spray when the

sprinkler head is given the coordinates (0, 0):

a) (9, 4) b) (8, 5) c) (8, 6) d) (9, 6)

Some students might do this using the graphing calculator approach, while others may

draw diagrams with compasses or use the distance formula.

It might help to have students focus on exactly what it means to have a circle with

radius 10 units. They should come up with a description of such a circle that generalizes to "A circle is the set of all those points in a plane that are a given distance

(the radius) from a given point (the centre)".

Using the distance formula, students should determine if specific points are within the

boundaries of the circle with radius 10. Ask students what equation could be written using coordinate variables x andy, so that the graph of the equation is a circle.

Students should test whether the point P (8, 6) lies on the circle using the distance formula:

OP = .Jcs-0)2 +(6-0/ = .Jsz +62 = .J64+36 = JlOO = 10

When (x, y) is used to replace (8,6), the equation describes any point on the circle.

0~ =~(x-0)2 +(y-Of =~x2 + /

They know the radius (OP) is 10, so:

The equation of a circle with radius 10 can be expressed as :x? + y = 102• Emphasize

with the students, that this equation produces a circle. Students can now conjecture

that

1) If the coordinates of a point satisfy the equation, then the point is on the circle. 2) If a point is on the circle, then its coordinates satisfy the equation.

This is another opportunity to use the term converse.


Circle Geometry

Worthwhile Tasks for Instruction and/ or Assessment

Pencil and Paper (E15/E4/D 1)

1) A ranger station, S, is located on the line 2x- 3y = -2. The area of sight from the station is bounded by the circle. a) Find the equation of the boundary. b) If a second station is located at T, find its

coordinates.

2) A circle is defined by :x? +I == 10.

y

a) Show that AB is a chord of the circle where A (1 , -3) and B (-3, 1). b) Find the equation of the right bisector of AB. c) Show that the right bisector of AB passes through the centre of the circle.

3) The equation of a circle is given by :x? + 1- 4x + my- 18 = 0. If A (7, 3) is a point on the circle, find the value of m.

4) Prove that the line x- 3y + 24 = 0 passes through the centre of the circle given by :x? +I+ 12x- 12y + 27 = 0 .

5) a) Graph the region defined by the inequalities

x2 + / - 2x + 4 y- 5 ~ 0 and x + y- 1 ~ 0.

b) Determine the area of the region defined in (a).


Suggested Resources

135

.)L.U: 1n unJ ~~-- - . e, students

will be expected to

El3 analyse and translate

between symbolic, graphical,

and written representations of

circles and ellipses

E3 write the equations of

circles and ellipses in

transformational form and as

mapping rules to visualize and

sketch graphs

El6 demonstrate the

transformational relationship

between the circle and the

ellipse

El4 translate between

different forms of the

equations of circles and

ellipses

El5 solve problems involving

the equations and

characteristics of circles and

ellipses




circles




arguments

136


El3/E3/El6 Reconsider the sprinkler system problem. Suppose the sprinkler head is

moved to a new location Q (3, -1). Keep the radius 10 and find the equation of the

circle. From their earlier work:

RQ =~(x-3Y +(y+ lf

===> (x- 3 Y + (y + 1 f = 102

From this centre-radius form of the equation, students can see the coordinates for the

centre of the circle, just as they would see the vertex for the parabola in a quadratic

function. This circle equation could be expressed in transformational form as

[ 1~(x-3)J +L~(y+1)J =1 where the stretch factors give the radius of the circle. As a

mapping rule the image equation can be expressed as (x.y) --7 (lOx+ 3,10 y -1) .

If the water pressure was turned up and the sprinkler head could now throw water 12

units, the equation would become [ 1~ (x- 3) J +L~ (y+ 1) J = 1

Extending the understanding of the equation of a circle to that of the equation of an

ellipse is a simple matter of understanding that the equation of the ellipse is

generated from the equation of a circle by having different vertical and horizontal

stretches. Its equation, in transformational form, would be

El4 Completing the square is used to manipulate equations of circles and ellipses

into transformational form, or in centre-radius form and equation of ellipses into

transformational form. For example, when students are asked to graph

:x?- + 2x + j- 4y = 12, they would write the following:

x 2 + 2x+ /-4y+ _ = 20

x 2 +2x+_+ /-4y+4= 20+1+4

( x + 1 )2 + (y- 2 )

2 = 25 (this is the centre-radius form)

[ i(x + 1) J + [ i(y- 2) J = 1 (this is the transformational form)

From this form they can describe the transformations of :x? + j = 1: "The centre (0,

0) is translated left 1, and up 2, and the radius is 5 units."

El5/E7 /Ell Students should solve problems using the equations for circles and

ellipses. For example: A circle is defined by :x?- + j- lOx -lOy+ 25 = 0. Show that

the line joining A (2, 1) to B (5, 0) is a chord of the circle, and that the right bisector

of AB passes through the centre of the circle.

ATLANTIC CANADA MATHEMATICS CURRICULU M: MATHEMATICS 3204/3205

Circle Geometry


Pencil and Paper (£3/E 16)

1) Find the equation of the image of XI+ j- = 1 under each mapping. Write the equation in standard form.

1 a) (x,y) -H3x-2,- y+ 1)

2

c) (x,y)-Hx+3,y-2)

b) (x,y) -H5x,y- 5)

d) (x,y) -HJ3x+ 5,J3y)

1 e) a vertical stretch of 2, a horizontal stretch of 3, centre (-2, 5)

2) What is the centre for the circle or ellipse defined by each equation above? If any of the above are circles, state the radius. If any are ellipses, state the length of the major and minor axis.

Pencil and Paper (E13/E3/E14)

3) Express each equation in transformational form. State the transformation of XI + j- = 1, and the mapping rules then sketch the graph.

a) x 2 +/-8x-9=0 b) x 2 + / -10x -6y -2=0

c) 2x2 + 2/ + 5 y = 0 d) 2x2 + 2/ - 4x + 6 y- 2 = 0

e) x 2 + / - 8x + 6 y -11 = 0

Pencil and Paper (E15)

\ 4) The dome of an arena is elliptical in shape. If the height of the dome is 28 m , and it has a span of75 m , find a possible equation for this ellipse.

5) An ellipse is given by the equation 25x2 + 4y2 + lOOx - 16y+ 16 = 0.

a) What are the coordinates of its centre? b) Define the mapping applied to the circle x?- + j- = 1 to obtain this ellipse.

6) The points P=( -1,~) and Q =(I, 8~ ) are on the ellipse :: + ; : . Find the values

of a2 and b2•

Pencil and Paper (E15/E7/Ell)

7) Given the circle and ellipse as in the graph a) Determine the equation for both the circle and the ellipse. b) Determine the length of the two chords. c) Determine the larger ratio:

i) the length of the longer chord to the centre of the ellipse ii) the length of the shorter chord to the centre of the circle


CIRCLE GEOMETRY

Suggested Resources

137

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to

C20(Adv) represent circles

using parametric equations

C36 demonstrate an


relationship between angle

rotation and the coordinates

of a rotating point

138


C20(Adv) Students should investigate how technology "draws" circles. For example, a

calculator calculates coordinates for points and so turns numbers into shapes on the

screen.

C36 Students should understand that if the cursor traced a

circle on the screen, it would report coordinates (x, y) along the

circle. To simulate this, students should draw a unit circle

(radius 1) with centre (0, 0). Try e = 30° , where the radius

meets the circle drop a perpendicular to the x-axis and calculate

the value of x, or the length from the origin to the intersection

X X

Point cose= --~ cos30° =-~ x = cos30° radius 1 ·

Similarly, sin 30° ~ f ~ y = sin 30° .

If 8 = 135° , the coordinate where the radius meets the circle is (cos 135", sin 135°).

Likewise, if e= 238° , the calculator would plot the point (cos 238°, sin 238°) or

approximately(- 0.53,- 0.85).

Also students should then be able to find the angle of rotation (e) given a coordinate

on the unit circle such as (.34, .94). They should approximate e to be about 70°.

C20(Adv) The angle e is called the parameter of x = cose, y =sine. It is a variable

used to describe another variable. Equations that contain parameters are called

parametric equations. So, if students wanted to draw a circle using parametric

equations, they would assign e the values oo to 360° and graph x = cose, y =sine.

On the TI-83 calculator, go to parametric mode and in the window, set

T = 0° ---? 360° , T-step at 5, x-values from -3 to 3, andy-values from -2 to 2 . Then at

''y =," students would enter x1 7

= cos ( T) and y 17

= sin ( T). Push Enter to graph a

circle with radius one.


I

Circle Geometry


Paper and Pencil (C36)

1) Without using trigonometric ratios, calculate the coordinate of the point Pas it rotates through the following degrees on a unit circle, centre (0, 0). Express your answer using exact values.

a) 45° e) 30° b) 135° f) 150° c) 225° g) 210° d)315° h)330°

i) 60° j) 120° k) 240° 1) 300°

2) Describe the patterns you see in the above results. 3) Find the same values in (1) above using trigonometric ratios and a calculator. 4) Find the values cos 50° and sin 50°. Explain the value of each in terms of a unit

circle. 5) On a unit circle, the coordinates of the image of (1, 0) after a rotation are

(-0.6282, 0 .7781). a) find the angle of rotation correct to two decimal places. b) What would be the arc measure from (-1, O) to (-0.6282, 0.7781)?

6) A circle, drawn on a coordinate plane, is centred at the origin with radius 25. If the terminal arm of an angle in standard position is (7, -24), what is the size of the angle?

7) Evaluate (Sin 240° r + (Cos 300° r Paper and Pencil ( C20(Adv))

8) Using parametric mode, write the equations that will produce a unit circle with centre (0, 0). State your 'window' settings.

9) How would you change your equations in # 6) above to increase the radius, and produce a circle with radius 5 units. State your window settings.


CIRCLE GEOMETRY

Suggested Resources

139

CIRLCE GEOMETRY

Circle Geometry

Outcomes


will be expected to

C37(Adv) describe and apply

parameter changes within parametric equations of circles

140


C37(Adv) Students should be encouraged to investigate all the parameters in the list

below:

• What happens as T-step is increased? Explain this behaviour. (Fewer points are being calculated, which because of the pixel display has a positive effect on the appearance of the circles up to t-step about 20).

• With x set to -6 to 6, andy to -4 to 4, investigate circles produced by 2 cos T, 2 sin T; 4cos T, 4 sin T, and explain how the "2" and "4" affect the circle (the "2" and "4" are the actual radii of the circles).

• Now explore the graphs of 2 + cos T, 2 + sin T; 4 + cos T, 4 + sin T (the "2" moves the centre to (2, 2), the "4" moves the centre (4, 4), and a small T-step is better). Also, a better window is needed-add two of each of the previous values.

• Explore adding a value to Tin the argument: cos (T + 5) , sin (T + 5); cos (T + 1), sin (T + 1). (Same circle-the " T ' value has been incremented, which does not affect the size or position of the circle.)

• Explore multiplications of Tin the argument: cos 2 T, sin 2 T; cos 5 T, sin 5 T (Again, this produces the same circle-affects T-step-not the position or size of the circle.)

In summary, when the variable Tis multiplied by numbers, the radius is affected, but when numbers are added to T, the location of the centre of the circle is transformed.

Students will want to justify their conjectures by making up examples, estimating

position and size, then graphing to check.

For example: 3cos8, 3sin8, 7 + 3cos8, 4+ 3sin8 .

Finally students should change the range ofT-step to various numbers less than 360° to see partial circles or pieces of circles.


Circle Geometry


Paper and Pencil ( C3 7 (Adv))

1) Write parametric equations to represent the circles in each of these graphs.

0 0 /'"'\ 0 '-V c

0 0

Performance ( C3 7 (Adv))

2) Explain how to get each of these graphs displayed on your screen.

c 3) This is a design created with parametric equations and a

graphing calculator. Create your own design or picture and record the equations, settings, and proper ranges to duplicate it.

4) a) Set maximum T = 180°; use the equations X= cos (T) and Y =sin (T) .

0 )

( ,/

c

b) Set maximum T = 360°; use the equations X= cos (0.5 T) andY= sin (0.5 T) . c) Set maximu m T = 90°; use the equations X= cos (27) and Y =sin (2 T). How do these different descriptions affect the way the graph in g calculator plots the points? Which is the fastest? Which is the most accurate? Explain your answer.


CIRCLE GEOMETRY

Suggested Resources

141

CIRLCE GEOMETRY


Unit 5 Probability (10- 15 o/o)

PROBABILITY

Probability

Outcomes


will be expected to

G2 demonstrate an

understanding that

determining probability

requires the quantifying of

outcomes

G3 demonstrate an


fundamental counting

principle and apply it to

calculate probabilities of

dependent and independent

events

144


G2 Every day students experience a variety of situations. Some involve making

decisions based on their previous knowledge of similar situations.

• Should they do their math homework tonight or during their spare period before math class tomorrow?

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

Before making the decision, what they must consider is, "What is the chance of this

decision working out in my favour?"

In probability, the goal is to assign numbers between 0 and 1 inclusive to events that

interest us, but for which we do not know the outcome.

In their previous studies (grades 7-9) students have created and solved problems

using probabilities, including the use of tree and area diagrams and simulations.

They have compared theoretical and experimental probabilities of both single and complementary events and dependent and independent events. They have examined

how to calculate complementary events as well as two independent events, A and B.

They have determined how to calculate the probability of A and Bas P (A )x P (B).

Sometimes the task of listing and counting all the outcomes in a given situation is

unrealistic, since the sample space may contain hundreds or thousands of outcomes.

G2/G3 The fundamental counting principle enables students to find the number of

outcomes without listing and counting each one. For independent events, if the

number of ways of choosing event A is n(A) and the number of ways of choosing

event B is n(B), then n(A and B)= n(A)xn (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 three

red marbles and two blue marbles, all the same size. A marble is drawn at random.

3 The probability that it is red is 5 . If the marble is then replaced, the probability of

3 picking a red marble again is 5 . However, if it is not replaced, then when another

2 marble is picked the probability of its being red is now 4 . The second selection of a

marble is dependent on the first selection not being returned to the box.

continued ...


Probability


Activity (G2/G3)

1) Two students are playing "grab" with a deck of special "grab" cards. One student has a triangular-shaped deck with 16 ones, 12 twos, 8 threes, and 4 fours. The other has a rectangular shaped deck with 10 each of ones, twos, threes, and fours. The decks are well shuffled and each student plays the top card simultaneously. 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

that could be played? b) How many of these are "double ones," that is, a one from the triangular deck

and a one from the rectangular deck? c) How many are i) double twos? ii) double threes? iii) 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 this principle and your answers to (c) to find the probability of i) a double

one ii) a double two iii) any double. g) 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

2) Telephone numbers are often used as random number generators. Assume that a computer randomly generates the last digit of a telephone number. What is the probability that the number is a) an 8 or 9? b) odd or under 4? c) odd or greater than 2?

3) A airplane holds 176 passengers: 35 seats are reserved for business class, including 15 aisle seats; 40 of the remaining seats are aisle seats. If a passenger arrives late and is randomly assigned a seat, find the probability of that person getting an aisle seat or one in the business section.

4) Use the given table, which represents the number of people who died from accidents by age group to find the following: [in each case assume that one person is selected at random from this group] a) the probability of selecting someone under 5 or

over 74 b) the probability of selecting someone between 16

and64

Age

0-4

5-14 15-24 25-44 45-64 65-774

75 and over

c) the probability of selecting someone under 45 or between 25 and 74


Number

3,843

4,226 19,975

27.201 14.733 8,499 16,800

PROBABILITY

Suggested Resources

Flewelling, Gaty et at.,

Mathematics 10 A Search for

Meaning. Toronto: Gage

1987.

145

PROBABILITY

Probability

Outcomes


will be expected to

G3 demonstrate an


fundamental counting

principle and apply it to

calculate probabilities of

dependent and independent

events

146


continued ...

G3 How is the fundamental counting principle related to probability? Consider the

marble situation described at the bottom of the previous page. The probability of

selecting red is P (c)=~ , while the probability of selecting blue isP (b)=~ (without

replacement) . The protability of selecting a red and a blue without replacgment

3 2 6 would be P(r and b)=-X- =-

5 4 20 .

Now, let us consider another situation:

Consider the experiment of a single toss of a standard die. There are six equally likely

outcomes: 1, 2, 3, 4 , 5 , and 6. Define certain events as follows:

A: observe a 2

B: observe a 6

C : observe an even number

D : observe a number less than 5.

1 1 P(A)=G (observe a 2), P(B)=G (observe a 6). What about P(A or B)

n(A) + n(B) 1 + 1 2 (observe a 2 or 6)? This can be shown two ways: total number of ways = 6 = 6

1 1 2 or P(A or B)= P(A)+ P(B)=6+6=6 . Will this be true for any two events? The

events "observe a 2 ", and "observe a 6" are called 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 C and D above have at two elements in common

and therefore are not mutually exclusive.

D~C 1 2 3 4 6

5

Consider the events C and D. The event (Cor D) includes all

the outcomes in C or D or both.

That is, P(C or D) = P(observe an even number or a number less than five)

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

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

5 outcomes. Thus P(C or D)=6

3 4 7 But P(C)+ P(D)= 6+6 = 6 , which cannot be possible since it exceeds 1.

The outcomes 2 and 4 are contained in both C and D and must be removed. There is

an overlap.

3 4 2 5 P(C or D)=P(C)+P(D)-P(C andD)=-+---=-6 6 6 6


Probability

Worthwhile Tasks for Instruction and/or Assessment Performance (G3)

1) Discuss whether the following pairs of events are mutually exclusive and whether they are independent. a) The weather is fine; I walk to work. b) I cut a deck of cards and have 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 in a hat and mixed, a) find the probabiliry of making one selection and getting a birthday in April or

October b) find the probability of making one selection that is the first day of a month or a

July date

3) A store owner has three student part-time employees who work independently 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. b) If the absenteeism rates are 2.5%, 3%, and 6% respectively for three different

employees, 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 if no defects show up when 3 are randomly selected. a) Find the probability of approval if the selected bolts are replaced, are 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 card from an ordinary deck of 52 playing cards. What is the probability that Mary will select either an ace or a diamond? Below is Fred's solution. Explain what Fred is thinking. Will his attempt lead to a correct answer? Explain .

Journal

6) Consider the table of experimental results.

4+ 13 17 P(ace or diamond) = ---s2 =

52 Seldane

Drowsiness I 70 No drowsiness 711

781

Comment on the following solution attempts.

Placebo Control Total 54 113 237

611 513 1835 665 626 2072

a) If one of the 2072 subjects is randomly selected, the probability of getting someone who took Seldane or a placebo is

781 + 665 = 1446 = 0.3489 2072 2072 4144

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

781 + 237 = 1018 = 0.491 2072 2072 2072


PROBABILITY

Suggested Resources

147

PROBABILITY

Probability

Outcomes


will be expected to

G4 apply area and tree

diagrams to interpret and

determine probabilities

148


G4 Students have studied area and tree diagrams since grade 7 and have applied them

to help establish the sample space, or the total number of possible outcomes in a

situation. In this course their experiences with these diagrams will be extended to

probability tree diagrams and area diagrams that will help students visualize and

calculate the probabilities given certain situations.

Consider the following situation. Students at Yore High School have two choices for

where to eat lunch, in the cafeteria or elsewhere outside the school. Mildred, the

manager of the cafeteria, needs to be able to predict how many students can be

expected to eat in the cafeteria over the long run. Mildred asks the math class to

conduct a survey. The results show that if a student eats in the cafeteria on a given

day, the probability that he or she will eat there the next day is 72%. If a student

does not eat in the cafeteria on a given day, the probability that he or she will eat in

the cafeteria the next day is 38%. On Monday, 80% of the students ate in the

cafeteria. What can Mildred expect for Tuesday?

A good way to organize all these statistics is with a probability tree diagram:

Monday Tuesday

C f . _2.1.1- Cafeteria Q.SO a etena -- Elsewhere

• ~ El h ~Cafeteria sew ere-- Elsewhere

Geometric or area models will be useful to some students as these models provide a

pictorial representation of the analysis which provides the students with a visual

insight into the concept of probability. Consider the following

situation. One of the events at your school's spring fair is a game of ~ chance involving points. For each turn, a player spins and gets the ~

points indicated in the area in that the spinner lands. Each player

should add the numbers obtained by spinning twice. What are all the

possible sums? What are the probabilities for obtaining each of

these sums?

Students will notice that the spinner suggests that -2 will happen

three-quarters of the time, while 5 will occur one quarter of the

time. Using a grid of 16 squares to represent

the probability of 1, they would draw a vertical line (as in fig. 1) to

represent the probabilities for the first spin (1/4 and 3/4). They

would then separate the grid horizontally (as in fig. 2) to represent

the probabilities of getting a -2 or 5 on the second spin. They

would then analyse the grid to find the probabilities of obtaining the sums --4, 3, and 10.

P ( -4) = :6

, P ( 3) = ( 1 ~) x 2, and P ( 10) =

1 ~ .

N j

spin 1

I ftgu rc I

10

3

3

3

spin 2 2

3 3

-4 -4

-4 -4

-4 -4

fig "" 2

3

-4

-4

-4

Now, using these results the students can be asked to create a situation where a player

must accomplish something in order to win the game.


Probability

Worthwhile Tasks for Instruction and/or Assessment Pencil and Paper ( G4)

1) This incomplete tree diagram lists all the outcomes of tossing

a coin and then rolling a die.

a) Copy and complete the diagram. b) How many pairs of outcomes are

there in this multiple event? c) What is the probability of tossing a head on the coin and then rolling a six on the die? d) What is the probability of tossing a head on the coin and then rolling an even

number? e) What is the probability of not tossing a head on the coin and then rolling an even

number of the die?

2) In a restaurant there are four kinds of soup, 12 entrees, six desserts, and three drinks. How many different four-course meals can a patron choose from? If 4 of the 12 entrees are chicken and two of the desserts involve cherries, what is the probability that someone will order wooton soup, a chicken dinner, a cherry dessert, and milk?

3) Licence plates for cars often have three letters of the alphabet, then three digits from 0 to 9. How many possible different licence plates can be produced? What is the probability of having the plate "CAR 000"?

Performance (G5)

4) The dart board at the right consists of four concentric circles whose centre is the centre of the square board. The side length of the square is 36 em. The circles have radii 2 em, 4 em, 6 em, and 8 em respectively. A dart hitting the hull 's eye or one of the shaded rings scores the indicated number of points. A hit anywhere else on the board scores 0 points. Assume that a dart thrown at random hits the board. Determine the probability of scoring: i) 4 points ii) 3 points iii) 2 points iv) 1 point v) 0 points

5) The following problem illustrates the usefulness of geometric probability. A tape recording is made of a meeting between a senator and her aide. Their conversation starts at the 2 1" minute on a 60-minute tape and lasts 8 minutes. While playing back the tape the aide accidentally erases 15 minutes of the tape. a) What is the probability that the entire conversation was erased? b) What is the probability that some part of the conversation was erased? c) Suppose the exact portion of the conversation on the tape is not known, except that it

began sometime after the 21 " minute. What is the probabiliry that the entire conversation was erased?

6) Consider finding the area of the region bounded by the ellipse 4.x2 + y = 4 . Enclose the ellipse in a rectangle whose sides pass through the x- andy-intercepts, and then consider the rectangular region to be a dart board. Suppose several darts thrown at random hit the rectangular region. a) Explain how probability can be used to approximate the area of the region bounded

by the ellipse. b) Explain how probability can be used to approximate the area of the region bounded

by the equation y = -.x-2 +4.

ATLANTIC CANADA MATHEM ATICS CURRICULUM : MATHEMATICS 3204/3205

PROBABILITY

Suggested Resources

149

PROBABILITY

Probability


will be expected to

G5(Adv) determine

conditional probabilities

150

Elaboration -Instructional Strategies/Suggestions G5(Adv) Ask the students a question such as What is the probability that event A occurs if it is known that event B has occurred? You should, through specific examples and some discussion, be able to get the class to arrive at a definition for conditional probability. For example:

If two dice, one red and one blue, are thrown and it is known that the blue die shows a number divisible by three, ask students what the probability is that the total on both dice is greater than 8? The condition that the number on the first die be divisible by three changes the sample space under consideration.

In particular, the new sample space contains only the 12 points shown inside the dashed closed curve at the right. In light of the

-- -----------6 :. • • • • • \ ... ____ ------- 1

(l) s . . . . . ·:a: ~ 4 ,· __ :_: __ • __ • _:,: ~ 3 I • e • e • • I

2 ":- -.--;-:- -.- -• , A

1 2 3 4 5 6 Red Die

fact that all36 points in the original sample space were assumed to be equally likely, students should agree that it seems reasonable to say that alll2 points in this sample space are equally likely. For how many of these points would the total be greater than 8? Given the condition that the number on the blue die is divisible by three, students should calculate the

5 probability of having a total greater than 8 is equal to U . For any two events A and B, the symbol " P (A jB) " is used to designate the probability that event A occurs given that event B has occurred. This is called a conditional probability because the condition is given that event B has occurred.

To evaluate P (A jB) reconsider the above problem. Let the original sample space be the set of 36 possible outcomes shown in the diagram, let A be the set of points for which the total number of spots showing is greater than 8, and let B be the set of points for which the number of spots showing on the first die is divisible by three. Then A II B , pronounced 'A intersect B' consists of the 5 points indicated in the diagram by the triangular shape. In this case, to determine the conditional probability P(A *B), divide the number of points in A II B by the number of points in B. Of course, if the points of the original sample space

were not equally likely, the result could not be obtained by simply counting points. Therefore, the probability of event A given that event B has occurred is defined as the probability of A 11 B divided by the probability of B.

The probability that event A occurs if it is known that event B has already occurred is known as "conditional probability." It is symbolized as P (A IB) , and calculated using

P (A IB ) = P (A n B) = 5 P (B) 12

If the first of three tosses of a fair coin is heads, find the probability of getting exactly two heads in three tosses.

Solution: Let A be the event "getting exactly two heads."

Let E be the event "getting a head on the first throw."

Theevent (AnE)={HHT,HTH}

2 1 1 so,P(AnE)=-=-

4, P(E)=-

8 2 1

:.P(AIE)= I =k 2

ATLANTIC CANADA MATHEMATICS CURRICULU M MATHEMATICS 3204/3205

Probability

Worthwhile Tasks for Instruction and/or Assessment Peiformance (G5(Adv))

1) What is the probability of getting two fives when two dice are thrown and it is known that at least one landed with a five up?

2) Assuming the probability ofbeing born male is 0.5. In a family of three children it is known that at least one child is male. What is the probability that all three children are male?

3) A weather report indicates an 80% probability of rain on Monday, 60% on Tuesday, and 20% on Wednesday. What is the probability that it will rain on at least one of the three days?

4) In the MAKE-A-NUMBER game, you draw a Condition Card. Then you draw two Number Cards from a stack of only five cards and place them side-by-side to

make a two-digit number. If the two-digit number fits your Condition Card, you score one point.

1. Condition

The number is divisible by 3.

Probability: ___ _

3. Condition

The number is greater thatn 40.

Probability: ___ _

5. Condition

The tens digit of the number is greater than the ones.

Probability: ___ _

2. Condition

The sum of the digits of the number is 5.

Probability: ___ _

4. Condition

The number is a prime number.

Probability: ___ _

6. Condition

The units digit of the number is divisible by the tens digit

Probability: ___ _

Determine the probability of scoring with these Condition Cards.

ATLANTIC CANADA MATHEMATICS CURRICU LUM: MATHEMATICS 3204/3205

PROBABILITY

Suggested Resources Shulte, Albert P. , ed. Teaching

Statistics and Probability. 1981

Yearbook. Reston, VA: NCTM, 1981.

151

PROBABILITY

Probability

Outcomes


will be expected to

G5(Adv) determine

conditional probabilities

152


G5(Adv) Tree diagrams are often used to organize all the possible combined outcomes

of a multiple event. Each student in a class of 30 students studies French or Italian

and one Science, Physics or Chemistry. Their choices are shown in the table.

Chemistry Physics

French 4 3

~talian 13 10

If a student is selected at random from the class, what is the probability that the

student studies French given that the student studies chemistry.

Number of students studying Chemistry

A Number who Number who selected French selected lralian

Num ber of students studying Physics

A Number who Number who studied French studied lcalian

Let F represent event "student studied French". Let C represent event "student

studied Chemistry" .

p (F!C)= P(Fnc) P(c)

4

17

An area diagram example: Suppose that Tom is a 60% free throw shooter in

basketball. At the end of a game he was fouled and his team is losing by two points.

He will shoot "one-and-one." What is the probability that he misses the second shot?

To solve this problem, students could use an area model like that on the right. The

probability of making the first shot is shown in fig. 1, then if he makes the first shot,

he gets the second shot. Fig. 2 shows the probability of missing the second shot.

-1.0-

i New sample ~ space

0.6 (1st shot made)

~ 0.4

~ fig. 1 fig . 2


Probability

Worthwhile Tasks for Instruction and/or Assessment Performance (GS(Adv))

1) Two gamblers play a game for a stake that goes to the first player to gain 10 points. If the game is stopped when the score is 9 to 8, in favour ofBill, what is the probability that Bill will win when the game is resumed? Use an area model to help. (It is assumed that both players have equal chances of winning each point.) If the score is 9-8 then rhe next score will be ...

EB -"'"'w'"' ;r r·~'l ~ ll':ri: ·· J 10-8, so- ' '·."' t . ' .

If the game goes to 9-9, either one might win. a) What can you conclude from this? b) What would be the solution to the problem if Bill was winning 9-7 when the game is

stopped?

2 1 2) As archers, Rita hits the target S of the time and David 3, of the rime. They are going

to have a contest with David shooting first. They alternate shots until one wins by hitting the target. Who is favoured? What is each contestant's probability of winning?

3) A certain restaurant offers select-your-own desserts. That is, a person may select one item from each of the categories listed: a) Using a tree diagram, list all possible desserts that can be ordered.

Ice Cream Sauce Extras

vanilla chocolate cherries

strawberry caramel peanuts

chocolate mint

b) Would you expect the choices of a dessert to be equally likely for most customers? c) If the probability of selecting chocolate ice cream is 40%, and vanilla is 10%,

chocolate sauce is 70%, and cherries 20% , describe the dessert with the highest probability ofbeing selected.

4) A certain model of automobile can be ordered with one of three engine sizes, with or without air 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 General American agency tells you there is only one of the cars on hand. What is the probability that it has the features you want, if you assume the outcomes to be equally likely?

5) Jennifer dresses in a skirt and a blouse by choosing one item from each category.

Skirts

tan plaid gray

stripe 1 stripe 2

stripe 3

Blo uses

white pink 1 pink 2

red

a) Show, by means of a tree diagram, all the outfits she can make if one has three striped skirrs and two pink blouses and only one of everything else.

b) What is the probability of her wearing something striped and white knowing that she already has a striped skirt on?

ATLANTIC CANADA MATHEMATICS CURRICULU M : MATHEMATICS 3204/3205

PROBABILITY

Suggested Resources

Newan, Claire et al. Exploring

Probability Quantitative

Literacy Series. White Plains,

NY: Dale Seymour

Publications, 1987.

153

PROBABILITY

Probability

Outcomes


will be expected to

G 1 develop and apply

simulations to solve problems

154


G 1 Simulation is a procedure developed for answering questions about real problems

by running experiments that closely resemble the real situation.

Suppose the students want to find the probability that a three-child family contains

exactly one girl. If students cannot compute the theoretical answer and do not have

the time to locate three-child families for observation, the best plan might be to

simulate the outcomes for three-child families. One way to accomplish this is to toss

coins to represent the three births. A head could represent the birth of a girl. Then,

observing exactly one head in a toss of three coins would be similar, in terms of

probability, to observing exactly one girl in a three-child family. Students could easily

toss the three coins many times to estimate the probability of seeing exactly one head.

The result gives them an estimate of the probability of seeing exactly one girl in a

three-child family. This is a simple problem to simulate, but the idea is very useful in

complex problems for which theoretical probabilities may be nearly impossible to

obtain.

Students need experience thinking through complete simulation processes. When

choosing a simple device to model the key components in the problem they have to

be careful to choose a model that generates outcomes with probabilities to match

those of the real situation. Students could use devices such as coins, dice, spinners,

objects in a bag, and random numbers.

Students need to understand that the experimental probability approaches the

theoretical probability as the number of trials increases. They should also realize that

knowing the probability of an event gives them no predicting power as to what the

outcome of the next trial will be. However, after enough trials, they should be able to

predict with some confidence what the overall results will be.

When conducting simulations students should follow a certain process such as the one

outlined: (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

Worthwhile Tasks for Instruction and/or Assessment Pencil and Paper ( G 1)

1) Consider the following problem: Marie has not studied for her history exam. She knows none of the answers on the sevenquestion 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 more of the seven questions. Ask students to complete the following: a) What are you 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 random points. What is the probability that the three pieces will form a triangle? (The pieces must touch end to

end. ) a) Describe the process that might be used to estimate the answer using experimental

probability. b) Instead, Robert is going to use a simulation. He assumes the spaghetti is 100 units

long, and he is going to generate two random numbers between 0 and 100 using each as a side of a triangle. How would Robert find the third side? How would Robert check to see if the numbers represent the lengths of the side of a triangle?

c) Perform this simulation to find the answer.

Performance (G 1)

3) Dale, a parachutist, jumps from an airplane and lands in a field. What are the chances that Dale will land in a particular numbered plot? Make a field grid using a normal 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-do not count it as a toss.) In your response consider several questions: Is there an equal chance to land in each plot? How many rimes did Dale land in plot 1?

~ rn

Discuss the experimental probability results versus the theoretical probability results for the given field.

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

c) Perform a simulation to answer the same problem as in a). Compare the results of the simulation with that of the theoretical. Comment.

4) Perform simulations to solve the following problems: a) What is the probability that all five children in a family will be

girls?

[B 6 5

b) A couple leaves for work anytime between 7:00 and 8:00am. Their newspaper arrives any time between 6:30 and 7:30am. What is the probability that they get the paper before they leave for work?


PROBABILITY

Suggested Resources

Zawojewski, Judith. Dealing

With Data and Chance.

Curriculum and Evaluation

Standards for School

Mathematics Addenda Series.

Grades 5-8. Reston, VA:

NCTM, 1992.

155

PROBABILITY

Probability

Outcomes


will be expected to

G7 distinguish between

situations that involve

permutations and

combinations

156


G7 Before describing different situations in terms of permutations and combinations,

students need to have an opportunity to solve simple counting problems (see

elaboration for G2, p. 142). They may wish to organize their work into systematic

lists and/or tree diagrams. As the number of choices increases, they will see the need

for a way to count more efficiently. For example:

a) How many different routes can you take from Sydney to Halifax through Antigonish?

b) How many routes are there from Antigonish to either Halifax or Sydney?

Following this, the class might be split into two groups-one will do Problem A, the

other Problem B. Students should present their solution to the class.

Problem A: Suppose there were three people, Adam, Marie, and Brian, standing in

line 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 many

ways can a committee of three people be formed?

Solutions might look like:

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 and

Elaine 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 the

same five people being the committee, so they represent one combination.

The essential difference between these two situations needs to be discussed and

emphasized. Eventually, Problem A should be described as a permutation (order is

important) , Problem Bas a combination (order not important) .


Probability


Pencil and Paper ( G7)

1) For each of the following, decide whether permutations or combinations are involved. a) The number of committees of two that can be formed from a group of 12

people. b) The number of possible lineups for a baseball team that can be formed from 12

people without regard to position (a baseball team consists of nine players, as follows: pi tcher; catcher; first, second, and third basemen; sho rtstop; right, centre, and left fielders).

c) The n umber of five-letter licence plates that can be formed from 12 different letters.

d) The number of six subsets that can be formed from 12 different letters. e) The number of five-man basketball teams that can be formed from 10 players. f) The number of ordered triples that can be formed from 10 different numbers. g) The number of ordered triples that can be formed from the numbers

1, 1, 1, 3, 3, 5, 5, 5, 5, and 4.

2) The manager of a baseball team needs to decide the batting o rder for the season opener. In how many ways can the first four batters be arranged on the batting roster? Is this a permutation or combination question? Explain .

3) As a promotion, a record store placed 12 tapes in one basket and 10 compact discs

in another. Pierre was the one millionth customer and was allowed to select 4 tapes and 4 compact discs. To find how many selections that can Pierre make, does one use permutations or combinations? Explain.

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 o rder?

RWRRW

5) a) Find the total number of arrangements of the letters of the word "SILK." b) Find the total number of arrangements of the letters of the word "SILL." c) How are your answers in a) and b) alike? How are they different?


PROBABILI1Y

Suggested Resources

157

PROBABILITY

Probability

Outcomes

SCO: In this course, students will be expected to

A 6 develop an understanding

of factorial notation and apply

it to calculating permutations

and combinations

GB develop and apply

formulas to evaluate

permutations and

combinations

158

Elaboration -Instructional Strategies/Suggestions A6 As students refine their methods of counting, moving from tree and area diagrams and listing through the fundamental counting principles, they should learn to recognize and use n! (n factorial) to represent the number of ways to arrange n distinct objects. For example, the product rule can be used to find the number of possible arrangements for three people standing in a line. There are three people to choose from for the front of the line. For each of these choices, there are two people to choose from for the second position in the line. For each of these choices, there is one person to choose from the end of the line. Therefore, there are 3 X 2 X 1 or six possible arrangements.

In another example, at a music festival, eight trumpet players competed in the Baroque class. After the judging, they were awarded first, second, third .. . down to eigth place. In how many ways could their placements be awarded?

If all the trumpet players were given a position first, second, third, .. . , eighth, then the total number of possible standings could be calculated by using reasoning like: There are eight people eligible for first, which leaves seven eligible for second, six people for third ... leading to a calculation 8 X 7 X 6 X 5 X 4 X 3 X 2 X 1 . This product can be written in a compact form as 8! and is read "eight factorial. "

In general, n! = n(n-l)(n- 2 ) ... (3 )(2)(1 ), where nEw and 0! = 1 .

A6/G8 If there are only three prizes to be given to the 8 trumpeters, how many ways could placement be awarded?

Students should reason that eight people could come first , only seven could come second, and six could come third --7 8 X 7 X 6 --7 336 . This could be worded "How many permutations are there of eight distinct objects taken three at a time?"

The symbol commonly used to represent this is 8P

3• or npr for the number of" n" objects

taken " r" at a time. Students should notice that

8 ~ = 8 X 7 X 6

8x7x6x5x4x3x2xl also, 8~ =

5x4x3x2xl 8!

so, 8~ = 5!

R = 8! 8 3 (8-3)!

n! so, n ~=(n-r)!

Students should note that when five people are to be arranged in a straight line there would be 5! or 120 ways to do this. However, if the same five people were to be arranged around a table in the order, say A, B, C , D , and E, their relat ive position to each other would not be distinguishable.

OA B OE A E C D B

D C Oc o

B E

A Thus, the total number of arrangements would be:


Probability


Pencil and Paper (A 6)

1) The town ofKarsville, which has 32 505 automobiles, is designing its own licence plates for residents 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 explain their choice: i) a licence made from using four single-digit numerals from 1 to 9 ii) a licence made of three single-digit numerals from 1 to 9 and one letter

from the alphabet iii) 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 and letters from the alphabet to suit the purposes of this town and to defend their selection.

2) In a box there are three black marbles and two white marbles.

Without looking in the box, choose two of the five marbles. How many ways are

there to select two marbles that are the same colour? Each a different colour?

Pencil and Paper (A6/G8)

a) Indicate which of the following are true (T) and which are false (F).

i)

ii)

iii)

iv)

5! -=5x 4 4!

10! 10 X 9 X 8 =-

7!

8 ~ = 56

100 1>;. = 100 X 100 X 99 X 98 X 97

b) Create a story where each true expression above would be used in the solution.

7) There are five points, no three of which are collinear, 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 four-ingredient 20 em pizza. If there are 15 ingredients from which to choose, how many different "specials" are possible?

9) Explain why the following theorem would be true:

p A circular arrangement of 'n' items can be calculated using: .!!.....!!.. = (n -1)!

n


PROBABILITY

Suggested Resources

159

PROBABILITY

Probability

Outcomes


will be expected to

A6 develop an understanding

of factorial notation and apply

it to calculating permutations

and combinations

G8 develop and apply

formulas to evaluate

permutations and

combinations

G7 distinguish between

situations that involve

permutations and

combinations

160

Elaboration -Instructional Strategies/Suggestions A6/G8 Refer back to the problem where there are five members on the executive of the student council. If these five were elected from a list of 10 candidates for executive position, such as president, vice president, secretary, the number of ways 10 people can be slotted into

10! five positions would be found using permutations toPs = (lO _ S)! 30 240

A6/G8/G7 From these five people a committee of three is struck. If the five people are represented by A, B, C, D, and E, then clearly a committee with A, B, and Cis the same as a committee with C, A, and B. So, the order of the selection is not important and the arrangement is called a combination. Therefore, since ABC, ACB, BAC, BCA, CAB, and CBA are all considered the same committee, they represent only one committee of three selected from the five people. The number of permutations of A, B, and Cis 3!. Thus, the number of committees from the original list ofl 0 candidates

number of ways the executive was chosen =----------~----------------------

3! 30240

=---3!

s = 5040

Th . c tOR, 4 at IS t O 3 = - - =50 0 3!

and the number of committees from the five member executive selected would be

c = 51{ = 10 5 3 3!

A combination of" n" objects taken " r" at a time is any subset of size " r" taken from the " n"

objects. The number is denoted by (n) (read "n" choose "r"), or C. r n r

The number ( ~) can be evaluated by investigating the connection between permutations

and combinations.

For example: A committee of size 4 and a committee of size 3 are to be assigned from a . n! group of 10 people. How many ways can this be done if

Smce n~ = ( ) no person is assigned to both committees? Solution: First n-r

Thus, in general, (n) = n ~ r r!

I . C = n. ··n r r!(n-r)!

Committee tO C 4 = ( ~ Q) = 21 Q ways, and there are 6

people left for the second committee. Second committee

6 C3 = ( !) = 20 ways. Therefore the two committees

can be assigned 210 • 20 = 4200 ways. Note: If the smaller committee was selected first then

c:) (:) = 120 · 35 = 4200 ways.


Probability


Pencil and Paper (A6/G8/G7)

1) a) Which of the following will produce the number of greatest magnitude? (Use estimation first.) Which will produce the smallest ?

9! i) 6! iv) 3!+4 vii)

7!

9! 100! ii) 11! v)

2! viii) --

2!

15! 9! ix) ) 7! iii)-, vi) 4!- 3! X-

12. 2! 6!

b) Pick three of the above expressions and create a problem in which these symbols would be used in the solution.

Pencil and Paper (A6)

2) Write each as a ratio of factorials.

a) 7 X 6 X 5

b) 19x9x8x7x5x19

c) 1 0 X 9 X 8 X 7 X 6

Peiformance (G8/G7)

d)

e)

f)

30 X 29 X 29 X 12 X 11 X 10 X 9

20 X 19 X 18 X 17

50·49·48 · 47·46

5!

3) A government committee of size 9 is to be selected from five liberals, four reformers, and four new democrats. How many ways can this be done if each of the three parties must be equally represented?

4) Explain in words why you think a combination lock is called a combination lock instead of a permutation lock.

5) A fly goes from A to B in the grid by travelling only to the right or upwards. How many possible routes are there? How many routes are there that go through C, but not through D?

6) Linda, Gino, and Sam each draw 3 cards from a deck of 52 playing cards and do not replace them. a) If Linda goes first, in how many ways can she pick 3 cards? b) In how many ways can Gino draw his cards after Linda has drawn hers? c) Finally, in how many ways can Sam draw her cards?

7) A quarterback on a football team has seven different plays to use in a game. In order to confuse the defence of the other team, the quarterback does not want to repeat the same sequence of plays too often. How many different sequences of three plays has she to choose from if no play is repeated?

8) Mr. Burble teaches 182 students mathematics at Harry High. He tells his students that they must do these six problems, but that they can do them in any order. Is it possible for each of his students to do them in a different order? Explain.


PROBABILITY

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161

PROBABILITY

Probability

Outcomes


will be expected to

G I 0 connect Pascal's Triangle

with combinatorial

coefficients

G9 demonstrate an

understanding ofbinomial

expansion and its connection

to combinations

162

Elaboration -Instructional Strategies/Suggestions G 10 Students should be asked to take binomials like (x + y) and find simplified expressions for (x + y)2, (x + y)3, (x + y)4, etc., and look for patterns in their coefficients. They should be able to find a connection between the expansion power and that same row in the Pascal's Triangle

with respect to the coefficient values. ( ( x + y t) = 1 is the top row, row 0).

1 1 1

1 2 1 1 3 3 1

1 4 6 4 1 1 5 1 0 10 5 1

Pascal's Triang le

G9 The counting techniques, discussed in G3, p. 142, can be useful in the multiplying of polynomials. Looking at the product of (a+ b) (c +d)= ac +be+ ad+ bd, students should notice that each term in the expansion has one factor from (a+ b) and one factor from (c +d) . e.g., ac has two factors a and c. The a is from (a+ b) and the cis from (c +d). Thus the number of terms in the expansion is four since there are two choices from (a+ b) and two choices from (c +d). Students should also notice that since there are two factors (a+ b), and (c +d) there are two factors in each term of the expansion.

(a+ b )(c +d) --7 ac +be+ ad+ bd --7 4 (each term has two factors)

i two factors

in each term The product of one binomial and itself follows the same pattern.

(x + y )2

--7 (x + y )(x + y) --7 xx + xy + yx + Y.Y, but the multiplication would be completed by collecting the like terms and using exponents: x2 + 2xy + y2. Students should

consider (X + y r = (X + y) (X + y) .. . (X + y) --7 xxxxx + xx.xyy + ... + YY.Y.YY . Each term

is made up of five factors and using exponents will look like X' yh where

xxxxx --7 x 5 y -7 5 + 0 = 5

a + b = 5, e.g., xxxyy --7 x 4 y -7 x4'' --7 4 + 1 = 5

xxxyy --7 x 3 y 2 -7 3 + 2 = 5

G 10/G9 In collecting the like terms, how many terms will be made up of the two factors x2y3? To answer this students should count the number of ways to make xly, e.g., the two factors of x must come from two of the five factors in each term of (x + y)S . This can be done

(~) or 5C 2 = 10 ways. The three factors of y must come from the remaining three factors in

each term of (x + y) 5 and this can be done in only one way. So the coefficient of x?-y ( 5 C

3)

will be 10. Students should note that these coefficients are values in the fifth row of Pascals' Triangle.

Students should examine the pattern changes in the signs between terms when (x-y) 5 is expanded. Because the second term in the expression (x-y) 5 could be considered negative (-y), then the terms in the expansion that have odd numbers of y-factors will be negative. When exponents or coefficients are included in the binomial to be expanded (x?- + 3y)3 students should be aware that for every x-factor, there is now an x?--facror, and for every yfactor there is now a 3y-factor, e.g., whenx is replaced withx?- and ywith 3y the expansion becomes:


Probability


Paper and Pencil (G 1 O/G9)

1) What is the coefficient of the x)? term in each of the following?

a) (x + y) 6

b) (x- 2y) 6

c) (2x + y) 6

d) (3x- 2y)6

2) When examining the terms from left to right, find the specified term in each

expansiOn. a) 10th in (x- y) 12

b) 20th in (2x-1) 19

c) 8 thin (a+ b) 10

d) 2 nd in (.0 - 5)7 e) 3 rd in (I - 2x) 9

f) 15th in (1 + a 2)

24

3) a) Find the sum of the elements in each row, for the first six rows of Pascal's Triangle.

b) Find the number of subsets in a 0-, 1-, 2-, 3-, 4-, and 5- element set. c) How are parts (a) and (b) related? d) How many elements are there in an n-element set?

4) Find a decimal approximation for 1.0210 by writing it as (I + 0.02) 10 and calculating the first five terms of the resulting binomial series.

journal(GI OIG9)

5) Betty Lou missed math class today. Helen phoned her at night to tell her about how combinations are helpful when expanding binomials. Write a paragraph or two about what Helen would have told her.

6) When expanding (a2- 2b) 5, Wally gets confused about the exponents in his answer.

Write a paragraph to Wally to help him remember how to record the exponents on this expansion.


PROBABILITY

Suggested Resources

163

PROBABILITY

Probability

Outcomes


will be expected to

G 11 (Adv) connect binomial

expansions, combinations,

and the probability of

binomial trials

G 1 develop and apply

simulations to solve problems

B8 determine probabilities

using permutations and

combinations

164

Elaboration -Instructional Strategies/Suggestions G 11 (Adv) Many experiments consist of more than two parts, and if these parts are independent of one another, students can use the concept of a product model or tree diagram to help them with their counting and probability calculations. For example, when

flipping a fair coin three times, a tree diagram ,., Flip 2nd Flip 3rd Flip

-Tails determines that the sample space has eight outcomes. <:::::: Heads --- Heads

The probability of any one of them being selected is 1 I Heads T~i;s -=::::: H-?,~1~s

( 1 )

3 1 or Heads -=::::: H{.~1~s

8. P (of any one of eight outcomes)= -2 8

. Tails <:::::: or --- Heads Tails - Tails

G 1 /G 11 (Adv)/B8 The tossing of three coins discussed above could be used as a simulation model. Say the students want to solve the problem "in a family with three children what is the probability that the first two children are girls and the third is a boy."

Experiments that consist of repeated trials of a simple experiment (e.g. , tossing a coin) using a model with only two possible outcomes (heads or tails) are called binomial trials. Suppose students needed to find the probability of getting exactly three heads in 10 tosses of a fair coin. Since each trial has two choices there would have to be 2 10 = 1024 branches on a tree diagram. The answer would be the sum of all the probabilities of the branches that contain three heads and seven tails. The number of ways 3 heads and 7 tails could be arranged is the

"ten choo•e th<ee ('" C, ) " m c:) , and w the probabili ry of rhi' happening would be

#of success _ 10 C3

rota! nwnber of outcomes -y ·

Another way to consider this is tha1t the probability of ~ny one of the 2

10(btan)

3 c(h;s)~eing

selected would be the product of 2 for every H and 2 for every Tor 2 2 . Since

every branch with three heads and seven tails has the same probability die answer is the number of these branches times the probability for each branch. The number of branches

will be the n urn bet of way• of choo•ing the th <ee head, out of ten to"e' c:) . Hence,

(10)( 1 )3

( 1 )7

(

10J (10) 13 17

3 2 2 . Students should compare ~ and 3 2 2 to see how one is the

same as the other.

In general, in binomial trials there are two outcomes for each of n trials . One of the two outcomes is a "success," the other a "failure. " These are labelled p and q respectfully and

q = 1- p . The number of successes inn trials is labelled s. Thus the probability of getting s

•ucce"e' and n _,failure• inn binomial trial. i• (: )p' q•-•. For example, •uppo•e ,ruden"

conduct an experiment of flipping a coin. The com 1s bent, so the probability of heads

(success) is 0.3. If they flip the coin five times, what is the probability of(t~~ree tails2

and cv:o

heads? Students should now be able to answer this with P(3T,2H) = 2 (0.3) (0.7) .

Students should recognize this as a term in the binomial series that comes rom expanding (0.7 + 0.3) 5• Students might want to use the ' randBin' feature on their calculators, or other software technology to conduct experiments or simulations where random samples are needed from populations that include only two possible o utcomes (e.g., yeses, and nos).


Probability


Performance (Gll (Adv)/Gl )

1) Find the probability of getting 10 heads in 15 throws of a bent coin if the

2 probability of heads on the bent coin is 3.

2) Find the probability of getting exactly two ones in six rolls of a fair die.

3) If n = 4 and p = ~, for what value of swill [: )p' q"-' be largest? Answer the same

1 2 question for n = 4 andp = 3and for n = 5 and p = 3 .

4 4) If Jamie is serving he wins a tennis game against Sam with probability 5, but if he

2 is receiving he wins with probability 5. Jamie and Sam agree to play five games,

and Jamie bets that he can win two in a row. If]amie wins the toss, should he elect to serve or receive? Draw two tree diagrams and verifY your answer.

5) A teacher made up a fair 1 0-item true and false test. Kira missed a few days just before the test and thought if she answered the questions randomly selecting Ts and Fs, she might do allright. When she was done, she had 4 Ts and 6 Fs . What is the probability that Kira's 4 Ts and 6 Fs are correct? Show how to find the answer two ways.


PROBABILITY

Suggested Resources

165

PROBABILITY

Probability

Outcomes


will be expected to

G12(Adv) demonstrate an

understanding of and solve

problems using random

variables and binomial

distributions

BS determine probabilities

using permutations and

combinations

166


G 12(Adv)/B8 Once students have developed the pattern described on the previous

page ( ~ J p' q"-' , they can use it to calculate the probabilities of other related events:

For example:

Let x = number of times the bent coin is "heads" in five flips . Let P (x) = probability that it is "heads" x times. Therefore, with the probability of heads being 0.30 .. .

no head: P(0)= ( ~(0.3° )(0 .7 5 )= 0.16807 J

one head: P(1) = (~} 0.31 )(0.74) = 0.36015

As a check on the answers, students

should realize that x is certain to take on

one of the values 0 through 5. So

P (0 or 1 or 2 or 3 or 4 or 5) must equal

1 or 100%.

five heads: P(5) = ( ~}0.35 )(0.7° ) = 0.00243

The independent variable xis called a random variable since

you cannot be sure what value x will have on any one run of

the random experiments. The dependent variable P(x) is the

probability that the value is x . So Pis a function of a

random variable. The graph of P(x) for the above situation

is shown.

The function P shows how the total probability, 1.00000, is

P(x)

0 .4

2 3 4 5

"distributed" among the possible values of x . This function of a random variable is

often called a probability distribution. Since this particular distribution has

probabilities that are terms of a binomial series, it is called binomial distribution. It is skewed left since the probability of heads is only 0.3.

Binomial distributions occur when students perform a random experiment repeatedly,

and each time there are only two possible outcomes (e.g. , heads or tails, boy or girl,

win or lose, yes or no). Students have already learned that a normal distribution is the

result of recorded measurement of the same phenomena repeated over and over and

over again. Since the binomial distribution is the result of a very similar action or fact

repeated over and over and over again, it would be expected that it too would

approach a normal distribution if the given probability is 0.5. This can be simulated

quickly using the ' randBin' feature of the graphing calculator or other software

technology. For example: ' randBin' (10,0.5 , 10)~ ~.Once they have found the

probability distribution, they can use the properties of probability to calculate the probabilities of related events.

For example, if the bent coin is flipped five times, as above, then the probability of

getting at least two heads is P(x ~ 2) = P(2)+ P(3)+ P( 4)+ P(5).

= 0.3087 + 0.1323+ 0.02835+ 0.00243


Probability


Performance (G12(Adv))

1) Hered ity Problem: If a dark-haired mother and a dark-haired father have a recessive gene for light hair, there is a probabiliry of them having a light-haired baby. For this to happen, each must have a large x (dark hair) and a small x (light hair) gene. In order for the baby to be light-haired, it must have two small x genes a) What is their probabiliry of having a dark-haired baby? b) If they have three babies, calculate P (0), P (1 ), P (2), and P (3) , the

probabilities of having exactly 0, 1, 2, and 3 dark-haired babies, respectively. c) Show that your answers to part b are reasonable by finding their sum. d) Plot the graph of the probabiliry distribution, P.

2) Multip le Ch oice Test Problem: A short multiple choice test has four questions. Each question has five choices, exactly one of which is right. Willie Makin has not studied for the test, so he guesses at random. a) What is his probabiliry of guessing any one answer right? Wrong? b) Calculate his probabilities of guessing 0 , 1, 2, 3, and 4 answers right. c) Perform a calculation that shows your answer to part b is reasonable. d) Plot the graph of the probabiliry distribution in part b. e) Willie passes the test if he gets at least three answers right. What is his

probabiliry of passing?

4) What is the probabiliry of getting exactly 50 heads when 100 coins are tossed?


PROBABILITY

Suggested Resources

167

PROBABILITY

Probability

Outcomes


will be expected to

G 11 (Adv) connect binomial

expansions, combinations,

and the probability of

binomial trials

G 12(Adv) demonstrate an

understanding of and solve

problems using random

variables and binomial

distributions

B8 determine probabilities using permutations and

combinations

168


G 11 (Adv)/G 12(Adv) /B8 Using the ideas developed over the last two two-page

spreads, students can investigate some of the claims typically made in television and

newspaper advertising. For example, a television commercial states that 8 out of 10

cats prefer Purrfect Chow. The claim is based upon a particular test in which 8 out of

10 cats chose Purrfect when given a choice between it and another cat food. A

complaint is made by a rival cat food manufacturer. They say that 8 out of 10 would

not be unusual, if it is assumed that cats have no particular preference for Purrfect.

Assuming that cats will choose equally between one food or another randomly, what is

the probability of them choosing Purrfect, and what does this mean with respect to the claim made by the other manufacturer?

In their solution attempts, students could use the binomial model to calculate the

probability that exactly 8 of 10 chose Purrfect:

( 10)( 1)8

( 1 )2

( 1 )( 1) 45 . 8 2 2 = 45

256 4 = 1024 = 0

•0439

IfR is the number choosing Purrfect, then the full probability distribution would be:

r 0 I 2 3 4

(~) H~r-I 10 45 12 0 2 10 --

10 24 10 24 1024 10 24 10 24

The graph of this distribution looks quite normal, since the probability is 0.5 that

cats will choose Purrfect over the other food choice.

From the graph the result "8 or more out

of 1 0" is likely to occur in only about 5% of all samples of 10 cats. Based on

5 6 7 8 9 10

2 52 210 120 45 10 I --1024 10 24 10 24 1024 10 24 10 24

0.25

0.1875

0.125

0.0625

2 3 4 5 6 7 8 9 10

previous study 5% is not very likely and suggests that the assumption made by the

rival manufacturers is probably wrong. It appears likely that more than 50% of cats would indeed choose Purrfect.

The probability of 8 or more of the cats choosing Purrfect can be calculated using:

45 10 1 56 . P(8 or more)= --+ - -+--= - -=0.055

1024 1024 1024 1024 .


Probability


Peiformance ( G 11 (Adv) /G 12(Adv))

1) Eigh teen-Wheeler P roblem: Large tractor-trailer trucks usually have 18 tires. Suppose that the probability of any one tire blowing out on a cross-country trip is 0.03.

2 4 4 4 4

Ask students the following: a) What is the probability that any one tire does nor blow out? b) What is the probability that

i) none of the 18 tires blows out? ii) exactly one tire blows out? iii) exactly two tires blow out? iv) more than two tires blow out?

c) If the trucker wants to have a 95% probability of making the trip without a blowout, what must the reliability of each tire be? That is, what is the probability that any one tire will blow out?

2) Sally claims that she can predict which way a coin will land, either heads or tails. Tommy throws the coin eight rimes and Sally gets it right six rimes. Ask students to calculate, on the basis of a binomial model, the probability of a) getting six coin tosses correct out of eight b) getting six or more coin tosses correct out of eight c) Ask students if they think the result supports her claim? Explain your answer.

6) A blind taste test is organized to see if people can tell the difference between two different brands of orange juice. They have 10 "tastes". After each taste they have to say whether it is juice A or juice B. Ask students how often they would expect the participants to get it right before they were reasonably convinced that they could actually tell the difference.

7) A list of people eligible for jury duty contains about 40% women. A judge is responsible for selecting six jurors from this list. a) If the judge's selection is made at random, what is the probability that three of

the six jurors will be women? b) Prepare a probability distribution table and graph for the number of women

among the six jurors. c) The judge's selection includes only one woman. Ask students if they think this

is sufficient reason to suspect the judge of discrimination? Explain.


PROBABILITY

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PROBABI L11Y "lillili~~~~~~~~~~~~~~~~ ~~~~~mlllli~~~il" - 31162012887924 -------

-, ~ T -- · • •r

170 ATLANTIC CANADA MATHEMATICS CURRICULUM : MATHEMATICS 3204/3205

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Mathematics - Memorial University of Newfoundlandcollections.mun.ca/PDFs/cmc_curr/Mathematics3204or3205.pdf · 2015-08-07 · Mathematics Mathematics 320413205 RJ.!I IJ.'II GOVERNMENT