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Grade 12 Applied

Mathematics (40S)

A Course for Independent Study

Grade 12

applied mathematics (40s)

A Course for

Independent Study

2007Manitoba Education, Citizenship and Youth

Manitoba Education, Citizenship and Youth Cataloguing in Publication Data

510 Grade 12 applied mathematics (40S) : a course for

independent study

Previously published as : Senior 4 applied

mathematics (40S) : a course for distance learning.

ISBN-13: 978-0-7711-3895-9

1. Mathematics—Programmed instruction.

2. Mathematics—Study and teaching (Secondary). I.

Manitoba. Manitoba Education, Citizenship and Youth.

II. Title: Senior 4 applied mathematics (40S) : a course

for distance learning

Copyright © 2007, the Government of Manitoba, represented by the Minister of Education,

Citizenship and Youth.

Manitoba Education, Citizenship and Youth

School Programs Division

1970 Ness Avenue

Winnipeg, Manitoba, Canada R3J 0Y9

Every effort has been made to acknowledge original sources and to comply with copyright

law. If cases are identified where this has not been done, please notify Manitoba Education,

Citizenship and Youth. Errors or omissions will be corrected in a future edition. Sincere

thanks to the authors and publishers who allowed their original material to be used.

This document was published in 2002 as Senior 4 Applied Mathematics (40S): A Course

for Distance Learning.

Acknowledgements

Manitoba Education, Citizenship and Youth gratefully acknowledges the contributions

of the following individuals in the development of Grade 12 Applied Mathematics (40S):

A Course for Independent Study.

Course Writer

Harland Garinger Hapnot Collegiate Flin Flon S.D. No. 46

Course Editor

Kathy Innes Winnipeg, Manitoba

Members of the Development Team

Cam Bennet Dauphin Regional Dauphin Ochre Area No. 1

Comprehensive Secondary S.D. No. 33

Harland Garinger Hapnot Collegiate Flin Flon S.D. No. 46

Robert Haynes Red River College

Jim Hoger Treherne Collegiate Tiger Hills S.D. No. 29

Kathy Innes Kelvin High School Winnipeg S.D. No. 1

Steve Khan St. Norbert Collegiate Seine River S.D. No. 14

Eric MacPherson Faculty of Education University of Manitoba

Fred Pauls Mennonite Brethren Independent Schools

Collegiate Institute

Irvin Peters Garden Valley Collegiate Garden Valley S.D. No. 26

Roy Spivak Grant Park High School Winnipeg S.D. No. 1

Mila Stout Swan Valley Regional

Secondary Swan Valley S.D. No. 35

Katherine Tetlock Distance Learning and Manitoba Education,

Information Technologies Unit Training and Youth

Wayne Watt Curriculum Unit Manitoba Education,

Training and Youth

Grade 12 Applied Mathematics Acknowledgements iii

DPS 10-2007

Manitoba Education, Training and Youth Staff

School Programs Division

Lee-Ila Bothe Coordinator Production Support Unit

Program Development Branch

Carole Bilyk Consultant Curriculum Unit


Paul Cuthbert Project Manager Distance Learning and

Information Technologies Unit


Mark Gavard Independent Study Distance Learning and

Program Coordinator Information Technologies Unit


Grant Moore Publications Editor Production Support Unit


Katherine Tetlock Project Leader Distance Learning and

(until August 2000) Information Technologies Unit


Lindsay Walker Desktop Publisher Production Support Unit


Wayne Watt Consultant Curriculum Unit


iv Acknowledgements Grade 12 Applied Mathematics

Contents

Acknowledgements iii

Introduction ix

Module 1: Matrices

Introduction 3

Lesson 1 Definition of a Matrix 5

Lesson 2 Addition and Subtraction of

Matrices 13

Lesson 3 Multiplication of a Matrix by a

Scalar 19

Lesson 4 Multiplication of Matrices 23

Lesson 5 Performing Calculations with

Matrices Using Technology 29

Lesson 6 Applications of Matrices 37

Project 53

Answer Key Module 1

Module 2: Vectors

Introduction 3

Lesson 1 Vector Terminology 5

Lesson 2 Ruler and Protractor Scale

Diagrams of Vectors 11

Lesson 3 Introduction to Geometer’s

Sketchpad and Euklid 21

Lesson 4 Addition of Vectors to Find a

Resultant Vector 35

Lesson 5 Using Technology to Find a

Resultant Vector 41

Lesson 6 Applications of Vectors 45

Project 51

Answer Key Module 2

Grade 12 Applied Mathematics Contents v

Module 3: Design and Measurement

Introduction 3

Lesson 1 Using a Spreadsheet to

Calculate Perimeter, Area,

Surface Area, and Volume of

Geometric Shapes 5

Lesson 2 Finding the Costs of a

Construction that Involves the

Calculation of Perimeter, Area,

Surface Area, and Volume 21

Lesson 3 Design a Construction that

Satisfies a Specified Budget 29

Lesson 4 Using Models to Estimate

Solutions to Complex

Measurement Problems 35

Project 39

Answer Key Module 3

Module 4: Probability

Introduction 3

Lesson 1 Definition of Probability 5

Lesson 2 Counting Techniques Using

Diagrams 11

Lesson 3 Using the Fundamental

Counting Principle 19

Lesson 4 Probability Using Sample

Spaces 27

Lesson 5 Addition Rules of Probability —

Mutually Exclusive and

Inclusive Events 35

Lesson 6 Multiplication Rules of

Probability — Independent and

Dependent Events 41

Answer Key Module 4

vi Contents Grade 12 Applied Mathematics

Module 5: Periodic Functions

Introduction 3

Lesson 1 Creating a Sinusoidal Periodic

Function from Data 5

Lesson 2 Properties of a Sinusoidal

Periodic Function 15

Lesson 3 Finding an Equation of a

Periodic Function Given its

Graph 33

Lesson 4 Applications of Periodic

Functions 45

Project 63

Answer Key Module 5

Module 6: Personal Finance

Introduction 3

Lesson 1 Review Budgeting and Create a

Personal Finance Portfolio 5

Lesson 2 Renting or Buying a Home 13

Lesson 3 Net Worth Statements 35

Lesson 4 Buying or Leasing a Vehicle 43

Lesson 5 Investing Money for the

Future 49

Lesson 6 Filing Income Tax Returns 53

Answer Key Module 6

Module 7: Sequences

Introduction 3

Lesson 1 Sequences and Spreadsheets 5

Lesson 2 Using Spreadsheets to Solve

Applications of Sequences 13

Lesson 3 Developing Fractal Patterns

Using Geometer’s Sketchpad

or Euklid 29

Grade 12 Applied Mathematics Contents vii

Lesson 4 Using Sequences to Determine

Perimeter and Area of

Fractals 35

Project 41

Answer Key Module 7

Module 8: Statistics

Introduction 3

Lesson 1 Review of Sampling

Techniques 5

Lesson 2 Calculating Measures of

Central Tendency and

Measures of Dispersion 9

Lesson 3 Using Technology to Calculate

Measures of Central Tendency

and Measures of Dispersion 19

Lesson 4 Standardized Scores —

The Z-Score 25

Lesson 5 The Normal Distribution 31

Lesson 6 Z-Scores and The Normal

Distribution 39

Lesson 7 The Binomial Distribution 51

Lesson 8 Confidence Intervals for the

Proportion in a Binomial

Distribution 61

Answer Key Module 8

Self-Tests

Self-Test Answer Keys

viii Contents Grade 12 Applied Mathematics

Grade 12 Applied Mathematics

Introduction

Welcome to Grade 12 Applied Mathematics (40S): A Course for

Independent Study offered through Manitoba Education,

Citizenship and Youth.

As a student in a course for distance learning, you have taken

on a dual role — that of a student and a teacher. As a student,

you are responsible for mastering the lessons and completing

the exercises assigned at the end of each lesson. As a teacher,

you are responsible for checking your work carefully and noting

the nature of your errors. Finally, you must work diligently to

overcome your difficulties.

You should seek out a study partner for this course. Most

students find that a study partner helps them get through the

course with greater success. This study partner can help you

correct your assignments and module self-tests, as well as help

you prepare for the examinations. It will also be necessary from

time to time to test some of your work on a partner. This does

not always have to be the same person, and at times you may

want more than one test person. You will also be performing

mathematical experiments and gathering data in some modules

of the course. A partner could be very vital to your success in

these assignments.

The applied mathematics curriculum has been developed in

response to changing mathematical requirements. These

requirements have changed because of the increased use of

technology in everyday life, post-secondary education, and the

workplace. Business and industry require responsible

independent learners who are:

• able to communicate mathematical ideas

• flexible

• capable of teamwork

Grade 12 Applied Mathematics Introduction ix

• computer literate

• skilled in problem-solving techniques

• self-reliant

In Grade 12 Applied Mathematics, students will master

essential skills in topics that have become important in post-

secondary institutions, technology-based industries, and daily

living. You will gain both desirable and required benefits by

taking Grade 12 Applied Mathematics.

It is not mandatory that you have taken Grade 10 Applied

Mathematics and Grade 11 Applied Mathematics, but it is

highly recommended. If this is impossible, it may be very

advantageous for you to go through at least the Spreadsheet

and Exploring Math Using Technology modules in Grade 10

Applied Mathematics and the Quadratic Functions and Personal

Finance modules in Grade 11 Applied Mathematics, as these

modules involve the use of technology. The knowledge gained

from the modules are part of the prerequisites for this course.

Even if you have done spreadsheets in another course, you may

want to go over the spreadsheet module from the Grade 10

Applied Mathematics program.

Course Description

The course is divided into eight modules. Each module contains

lessons, followed by assignments. It is recommended that you

complete all of the assigned exercises. Solutions are provided for

the exercises that are part of this package. These solutions are

found at the end of the module. Throughout the eight modules

there will be projects to be completed. They will be signified by

appropriate icons.

x Introduction Grade 12 Applied Mathematics

The eight modules are as follows:

Module 1: Matrices

Module 2: Vectors

Module 3: Design and Measurement

Module 4: Probability

Module 5: Periodic Functions

Module 6: Personal Finance

Module 7: Sequences

Module 8: Statistics

The table of contents outlines the major topics and sub-topics

found in this course. Every student enrolled in this program is

required to complete all of the eight modules. Each module ends

with a self-test. With the exception of the Design and

Measurement and Personal Finance modules, the module self-

tests include material from previous modules. This is done to

help you prepare for the examinations so that you will have the

chance to review material throughout the program on a

continuous basis.

Self-tests should be written without the aid of any books. Your

performance on these tests will give you an indication of how

well you understand the material. Your study partner could

help you in marking these tests.

Senior 4 Applied Mathematics Introduction xi

Evaluation

o Module 1 Project 5%

o Module 2 Self-Test 10%


Midterm Examination 20%


o Module 6 Self-Test 10%


Final Examination 40%

Total 100%

You are required to send a cover sheet with each of the

completed hand-in assignments. Cover sheets can be found on

the pages following this Introduction.

Additional Resources

You will need access to a computer and a number of software

programs in order to complete the assignments in this course.

One of them, Quicktax 2002 came with this course package. The

other software that you have to obtain includes:

1. Microsoft Excel: If you do not have access to it, you could use

a similar spreadsheet program.

2. Winmat: This program is available online as shareware. You

can download it from <http://math.exeter.edu/rparris>. If you

are unable to locate it there, try using a search engine.

3. Winstats: This program is also available online as shareware.

You can download it from <http://math.exeter.edu/rparris>. If

you are unable to locate it there, try using a search engine.

4. A program to do fractals, vectors, and sinusoidal data:

You can use either option 'a' or 'b' below. If you are attending

school or want to purchase software, go to option 'a'. If you

are not attending school and do not want to purchase the

software, go to option 'b'.

xii Introduction Grade 12 Applied Mathematics

Note:

Hand-in

assignments are to

be sent to

ISO Tutor/Marker

555 Main Street

Winkler, MB

R6W 1C4.

a) Geometer's Sketchpad: If you are attending a high school,

ask your ISO coordinator if this program is available for

you to use. If it is not, you will have to go to option 'b'

below or purchase the software from the Manitoba Text

Book Bureau at <http://www.mtbb.mb.ca/> (stock number

8764).

or

b) Euklid and a draw or paint program: If you do not have

access to Geometer's Sketchpad, you can use a combination

of Euklid and another program. Euklid can be downloaded

from <http://www.dynageo.com/eng/index.html>. If you are

unable to locate it there, try using a search engine. Besides

Euklid, you need to use either the drawing tools found in

Microsoft Word or another word processor, or Windows

Paint. Paint can be found by clicking on the 'Start' menu at

the bottom left-hand side of your screen, then moving the

arrow to 'Programs', then to 'Accessories'.

5. A program to do curves. You can use either 'a' or 'b' below.

If you are attending school or want to purchase software, go

to option 'a'. If you are not attending school and do not want

to purchase the software, go to option 'b'.

a) Zap-A-Graph: If you are attending high school, ask your

ISO coordinator if this program is available for you to use.

If it is not, you can either go to option 'b' or purchase it

from the Manitoba Text Book Bureau at

<http://www.mtbb.mb.ca/> (stock number 8763).

b) If you do not have access to Zap-A-Graph, you can use

Curve Expert, which completely replaces all the work you

would need to do with Zap-A-Graph. You can download

Curve Expert at

<http://www.ebicom.net~dhyams/cvxpt.htm>. If you are

unable to locate it there, try using a search engine.

It would be beneficial to have access to the Internet. A graphing

calculator such as a TI-83 or TI-83 Plus would also be beneficial.

Grade 12 Applied Mathematics Introduction xiii

Icon Guides

Icon guides have been placed inside the margins of the course to

identify a specific task. Each icon has a specific purpose to help

guide you.

The significance of each icon guide is described below.

Think about this idea.

Record your work on a standard-size audio tape.

Check the checklist to ensure you have completed all the

necessary work.

Mail your sequence work to your tutor/marker.

Assignment: Complete the following assignment.

Include Cover Sheet from the end of this introduction with

your hand-in materials.

Take note!

Spreadsheet application: Assignment is easier to complete

with a spreadsheet application.

Exam time: When this graphic appears, it is time to write an

examination.

xiv Introduction Grade 12 Applied Mathematics

Exam

Time!

•• ••

Include

Cover

Sheet

Grade 12 applied

mathematics (40s)

Module 1

Matrices

Module 1

Introduction

In this module, you will be introduced to the use of matrices. In

a world where technology has allowed access to abundant

amounts of information, a greater need for organization of this

information has evolved. One of the ways of organizing data is

through the use of matrices. Anywhere there is a need for

networking, matrices can be used in decision making. Examples

could include: telephone systems; railway, airplane, and bus

scheduling; and Internet communication. Matrices are used in

keeping records of inventory and cost cataloguing in business.

Security systems, ranging from complex systems found in

national security to simple security built into software

programs, may make use of matrices. In this module, you will

learn what matrices are, how to perform operations with them,

and how they may be used.

You will use a software program called Winmat to perform

calculations with matrices once you have learned the basics on

how the calculations are done manually. These calculations can

also be performed using a graphing calculator such as a TI-83 or

HP48G. You may want to explore the use of the graphing

calculator on your own; however, in this module the lessons and

assignment answer keys will be done using Winmat. Refer to

page xii of the course introduction to find out how to obtain a

copy of Winmat.

Outline

Lesson 1 Definition of a Matrix

Lesson 2 Addition and Subtraction of Matrices

Lesson 3 Multiplication of a Matrix by a Scalar

Lesson 4 Multiplication of Matrices

Grade 12 Applied Mathematics Module 1, Introduction 3

Lesson 5 Performing Operations with Matrices Using

Technology

Lesson 6 Applications of Matrices

4 Module 1, Introduction Grade 12 Applied Mathematics

Lesson 1

Definition of a Matrix

Objectives

In this lesson, you will learn

• what a matrix is

• the basic properties of a matrix

1. Definition

A matrix (plural: matrices) is a rectangle of related numbers

that represent data. A matrix is usually named by a capital

letter.

Example 1

Example 2

Example 3

Distance Matrix:

Brandon

Flin Flon

Saskatoon

Regina

Winnipeg

Br FF Sask Reg W

Grade 12 Applied Mathematics Module 1, Lesson 1 5

Example 4

NHL Eastern Conference Standings (Top Eight Teams).

2. Dimension of a matrix

The number of rows (horizontal) and the number of columns

(vertical) determine the order or dimension of the matrix.

Example 1

2 x 3 matrix since there are two rows and three columns

Example 2

1 x 2 matrix since there is one row and two columns

GP W L T PTS

New Jersey

Pittsburgh

Philadelphia

Boston

Washington

Buffalo

Montreal

Ottawa

6 Module 1, Lesson 1 Grade 12 Applied Mathematics

Example 3

8 x 5 matrix since there are eight rows and five columns

What is the meaning of the dimension of this matrix? In the

matrix S, the dimension indicates how many teams are

analyzed and how many categories are being used.

3. Row Matrix

If a matrix has a single row, it is called a row matrix.

Example 1

4. Column matrix

If a matrix has a single column, it is called a column matrix.

Example 1

5. Square matrix

If a matrix has the same number of rows as columns, it is called

a square matrix.

GP W L T PTS

New Jersey

Pittsburgh

Philadelphia

Boston

Washington

Buffalo

Montreal

Ottawa


Example 1

2 x 2 or a square matrix.

6. Definition and notation for cells

The intersection of each row and column in a matrix is called a

cell. Cells are named using the row number and column number

that they appear in.

Example 1

A11 = 2 A12 = 3 A23 = 1

Example 2

If the dimension is m x n, the cell names would be:


Example 3

In the matrix:

Cell S52 would indicate the number of wins that Washington

has in the standings.

Cell S45 would indicate the number of points that Boston has in

the standings.

7. Diagonal of a matrix

The main diagonal of a square matrix contains the cells that lie

on the diagonal of the matrix that starts in the upper left corner

of the matrix and ends in the bottom right corner.

Example 1

In matrix the main diagonal contains

cells E11 = 1, E22 = 0, and E33 = 8.

New Jersey

Pittsburgh

Philadelphia

Boston

Washington

Buffalo

Montreal

Ottawa

GP W L T PTS


Example 2

Note that in any square distance matrix like

the main diagonal cells are all 0s. This is due to the fact that all

places in the distance matrix are 0 km or 0 miles from

themselves.

Brandon

Flin Flon

Saskatoon

Regina

Winnipeg

Br FF Sask Reg W


Assignment

1. A software company distributes three types of graphing

programs: T61, T62, and T63. Below is a table and a

corresponding matrix that shows the number ordered by four

of its stores.

a. What is the dimension of matrix A?

b. What does the dimension mean?

c. What is the value of A23?

d. What does A31 indicate?

2. The standings in the American League West division of the

Major Leagues of baseball, where W indicates wins, L

indicates losses, PCT indicates percentage of games won, and

GBL indicates the number of games behind the leader, were

as follows:

a. Create matrix W that expresses these standings at this

time.

T61 T62 T63Store 1 10 20 20Store 2 30 20 35Store 3 20 30 10Store 4 30 10 10

W L PCT GBL

New York 62 41 .602 -

Toronto 59 48 .551 5

Boston 57 47 .548 5.5

Baltimore 46 58 .442 16.5

Tampa Bay 43 62 .410 20


DPS 10-2007

b. What is the dimension of matrix W?

c. What does the dimension mean?

d. What information does cell W32 indicate?

e. What information does cell W24 indicate?

f. What cell indicates the number of games behind the leader

for Tampa Bay?

g. What cell indicates New York’s wins?

3. a. Using a map of Manitoba, create a distance matrix D that

indicates the distances between the following places:

Dauphin, Morden, Portage, Steinbach, and Winkler

b. What type of matrix is D?

c. What value should be in cell D22?

d. Which cell indicates the distance between Morden and

Steinbach?

e. Which cell indicates the distance between Dauphin and

Portage?


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