Course Profiles
Public and Catholic District School Board Writing Partnerships
Mathematics
Course Profile
Mathematics of Data Management
Grade 12
University Preparation
MDM4U
 for teachers by teachers
This sample course of study was prepared for teachers to use in meeting local classroom
needs, as appropriate. This is not a mandated approach to the teaching of the course.
It may be used in its entirety, in part, or adapted.
Spring 2002
Course Profiles are professional development materials designed to help teachers implement the new
Grade 12 secondary school curriculum. These materials were created by writing partnerships of school
boards and subject associations. The development of these resources was funded by the Ontario Ministry
of Education. This document reflects the views of the developers and not necessarily those of the
Ministry. Permission is given to reproduce these materials for any purpose except profit. Teachers are
also encouraged to amend, revise, edit, cut, paste, and otherwise adapt this material for educational
purposes.
Any references in this document to particular commercial resources, learning materials, equipment, or
technology reflect only the opinions of the writers of this sample Course Profile, and do not reflect any
official endorsement by the Ministry of Education or by the Partnership of School Boards that supported
the production of the document.
© Queen’s Printer for Ontario, 2002
Acknowledgments
Public and Catholic District School Board Writing Teams – Mathematics of Data Management
Public District School Board Writing Team
Lead Board – Peel District School Board
Teresa Gonzalez, Superintendent of Program Services
Irene McEvoy, Mathematics Coordinator
Kaye Appleby, Project Manager
Writing Team
John Rodger, Peel District School Board
Kevin Spry, Upper Grand District School Board
Fran McLaren, Upper Grand District School Board
Reviewers
Shelley Wong, Peel District School Board
Ron Lewis, Rainbow District School Board
Eric Muller, Mathematics Department, Brock University
Barbara Canton, Limestone District School Board
Christine Suurtamm, Faculty of Education, York University
Catholic District School Board Writing Team
Lead Board – Ottawa-Carleton Catholic District School Board
Lucy Miller, Superintendent of Educational Programs
Tom Steinke, Project Manager
Writers
Carolyn Crosby, Catholic District School Board of Eastern Ontario (Lead Writer)
Kathy Pilon, Catholic District School Board of Eastern Ontario (Lead Writer)
Reviewers
Sandy Dobec, Ottawa-Carleton Catholic District School Board
Dr. Marilyn Kasian, Ottawa-Carleton Catholic District School Board
Joel Yan, Statistics Canada
Tom Steinke, Ottawa-Carleton Catholic District School Board
 Mathematics of Data Management – University Preparation
Course Overview
Mathematics of Data Management, MDM4U, Grade 12,
University Preparation
Policy Document: The Ontario Curriculum, Mathematics, Grades 11 and 12, 2000.
Prerequisite: Functions and Relations, Grade 11, University Preparation;
or Functions, Grade 11, University/College Preparation
Course Description
This course broadens students’ understanding of mathematics as it relates to managing information and
focuses on a culminating project throughout the course. Students will apply methods for organizing and
analysing large amounts of information; apply counting techniques, probability, and statistics in
modelling and solving problems; and carry out a culminating project that integrates the expectations of
the course and encourages perseverance and independence. Successful completion of MDM4U prepares
students for any undergraduate course in probability and statistics. Such courses are typically a
requirement for students in their second year of most four-year undergraduate programs in both the
sciences and humanities. In particular, students planning to pursue university programs in business,
social sciences, or the humanities will find this course of relevance.
Course Profile Design
The course is comprised of four strands: Organization of Data for Analysis; Counting and Probability;
Statistics; and Integration of the Techniques of Data Management. In Organization of Data for Analysis,
students look at finding and retrieving the data needed to answer significant questions. In addition,
students develop facility with the use of diagrams and matrices to model and solve problems. In Counting
and Probability, students have opportunities to solve counting problems, use counting techniques to
determine and interpret theoretical probabilities, and design and carry out simulations to estimate
probabilities. In Statistics, students acquire the tools to analyse data involving one variable and solve
problems involving normal distribution. Students describe the relationship between two variables and
assess the validity of statistics drawn from a variety of sources. In Integration of the Techniques of Data
Management, students complete a major project on a topic of their choosing which requires them to
integrate expectations of the course. Students present their projects to the class and critique the projects
of others.
How This Course Supports the Ontario Catholic School Graduate Expectations
This course encourages the Catholic learner to develop his/her God-given gifts and abilities and to
promote growth toward personal responsibility in preparation for a chosen career path. Throughout this
course, emphasis should be placed on moral, ethical, and realistic decision making in an effort to build
responsible citizenship. The classroom environment should instil a spirit of cooperation, rather than
competition, amongst students and should foster a collaborative sense of community. This course
provides many opportunities for students to work effectively as interdependent team members and to
acknowledge others for their opinions.
Course Notes
The last strand is the focal point of the Course Profile. The overall expectations of this strand are that
students will:
 carry out a culminating project on a topic or issue of significance that requires the integration and
application of the expectations of the course;
 present a project to an audience;
 critique the projects of others.
Page 1
 Mathematics of Data Management – University Preparation
The course needs to be structured with the expectations of this strand in mind. In this profile, the course
is organized into five units, some of which encompass expectations taken from different strands.
In Unit 1: Posing Questions with Data, students are introduced to problems that require the retrieval
and/or manipulation of data from large bodies of information. In particular, students will have the
opportunity to:
 view some sample data sets and pose questions about those data sets;
 given a question, determine/locate the data needed to answer the question;
 use technology to search, effectively, for the data needed to answer particular questions;
 create database or spreadsheet templates that facilitate the retrieval of data from large bodies of
information;
 critique the appropriateness of data sets which have been gathered to answer specific questions.
In Unit 2, Data Analysis, students acquire the tools to demonstrate an understanding of standard
techniques for collecting data; analyse data involving one variable using a variety of techniques; solve
problems involving the normal distribution; describe the relationship between two variables by
interpreting the correlation coefficient; and evaluate the validity of statistics drawn from a variety of
sources.
In Unit 3: Counting and Probability, students solve introductory counting problems using Venn diagrams,
as well as the additive and multiplicative counting principles; develop techniques for counting
permutations and combinations; determine theoretical probabilities using combinatorial techniques;
construct discrete probability distribution functions; and calculate expected values within the context of
an application. In addition, students determine probabilities using the binomial distribution; design and
carry out simulations to estimate probabilities, and assess the validity of simulation results by comparing
them with the theoretical probabilities.
In Unit 4: Additional Tools for Data Management, students solve problems involving complex
relationships with the aid of diagrams (e.g., network diagrams, tree diagrams, cause-and-effect diagrams,
timelines), and model situations and solve problems using matrices.
In Unit 5: Managing the Culminating Project, students prepare to successfully complete expectations
associated with the culminating project outlined in the strand, Integration of the Techniques of Data
Management. Students engage in several activities in which they apply several of the techniques/tools of
the course to answer significant questions. The activities are not clustered in a single chunk of time.
Instead, they are used at appropriate times in other units or near the end of related units.
The Grade 12 course profiles represent a collaborative effort between the Public and Catholic writing
teams. While not as detailed as previous profiles, they are designed to complement and supplement each
other. In addition to two complete “sample” units, a less-detailed Unit Overview chart offers a
recommended clustering of expectations for each of the remaining units, providing a starting point from
which teachers can develop their own, individualized units.
For some students, mathematics is perceived to be a collection of isolated and complex topics, each
requiring skills that may soon be forgotten. The mathematics teacher must address these perceptions by
creating a context in which students can learn and connect concepts and skills. Students must be exposed
to a variety of teaching, learning, and problem-solving techniques to best synthesize the information
presented by the curriculum and should be provided applications and context to bring meaning to their
learning.
Note: The activities in this profile both introduce and consolidate skills necessary for success in this
course. The activities can be used in conjunction with or independently of one another. Alternate
teaching strategies and technological tools are suggested.
Note: Teachers will note that some of the data sets provided here, and those that will be encountered on
available licenced software, will pertain to issues that can be sensitive or troublesome to some students.
Page 2
 Mathematics of Data Management – University Preparation
Teachers will need to carefully consider the selection of such items for student use, being sensitive to
individual student circumstances. The screen captures and graphics can be found within the software
program used in presenting the activities.
Because this course has been designed to prepare students for entry into various programs at university,
the specific nature of the learning activities should reflect this destination. In particular, students in this
course should routinely be challenged with problems and questions, which require investigation,
research, data collection, analysis, and reflection. Students must engage in activities that require
collaboration with other students and other activities that need sustained, independent effort. Students
must also have the tools and the time to work on complex tasks to develop their problem-solving skills.
Students with learning disabilities need specific guidance to benefit from the investigative approach
presented in this profile. Review of prerequisite skills and instructions in the use of technology, and in
particular graphing calculators, will be required before any activities are begun. Clear and precise
instructions with examples will need to be provided.
Several of the activities presented in this profile include extensions of the required content, which can be
used to meet the need to challenge gifted students. Other accommodations may include allowing for
student preferences in supplemental learning, altering the pace of instruction, creating a flexible
classroom environment, and using specific instructional strategies. Creative approaches to problem
solving must be encouraged.
The Achievement Chart for Mathematics is the basis of all assessment and evaluation for this course. The
Grade 10 Principles of Mathematics Academic Public Course Profile includes charts suggesting
strategies that can be used for the assessment and evaluation of all categories of the Achievement Chart
(p. 11). A chart outlining the component actions that are needed for successful inquiry and problem
solving is also included (p. 12). These charts provide an excellent base with which to begin the
implementation of these strategies, and for teachers of this course to extend, depending on their degree of
readiness. Another excellent resource is the Concerning Assessment and Reflective Evaluation (CARE)
package, available for free download at www.oame.on.ca. Included in this package are generic rubrics for
Communication and Thinking/Inquiry/Problem Solving Skills, along with suggested applications of these
instruments.
Units: Titles and Time
* Unit 1
Posing Questions With Data
Unit 2
Data Analysis
Unit 3
Counting and Probability
Unit 4
Additional Tools for Data Management
** Unit 5
Managing the Culminating Project
* Unit 1 is fully developed by the Catholic Course Profile writing team.
** Unit 5 is fully developed by the Public Course Profile writing team.
21 hours
23 hours
20 hours
20 hours
26 hours
Unit Overviews
Unit 1: Posing Questions With Data
Time: 21 hours
Ontario Catholic School Graduate Expectations: 1d, 2b, 2c, 3b, 3c, 3d, 3e, 4e, 4f, 5a, 5b, 5e, 7e, 7i
Unit Description
Students learn to find, retrieve, and organize credible data. They learn to pose significant questions
through the use of journals and critique the work of others. Some activities are grouped to teach the
expectations in an instructional activity followed by an assessment activity.
Page 3
 Mathematics of Data Management – University Preparation
Using Fathom, students locate and retrieve large data sets from a variety of Internet sites, including
Statistics Canada (E-STAT). Students answer questions using the data sets and consider and explore
other factors that could influence the data. They use the analysis features of Fathom to analyse one- and
two-variable data; analyses include cause-and-effect and regression. Students present their findings in
small-group settings and critique the data analyses of others clearly, honestly, and with sensitivity.
Students complete the unit by posing a problem, finding and analysing data, presenting their work on a
poster, and critiquing the work of others.
Unit Overview Chart
Activity
Expectations
Assessment
Tasks
1.1
DMV.02, DM2.02
Communication
A hook to begin looking
Posing Questions and
CGE5a, 7e
Teamwork
at and reading graphs and
Reading Graphs
Inquiry
a beginning point to
stimulate ideas for the
1 hour
culminating project.
1.2
ODV.01, OD1.01,
Knowledge
Locate and retrieve data
Introduction to E-STAT
OD1.02
Works Independently to answer questions.
(Instructional Activity)
CGE2b, 3c, 3d, 7i
Teamwork
3 hours
1.3
ODV.01, OD1.01,
Knowledge
Use E-STAT to locate
Posing Questions,
OD1.02, DMV.02,
Communication
and retrieve data to
Finding Data, and
DM2.01, DM2.02,
Application
answer questions; present
Critiquing Conclusions
DM2.03
Initiative
work and critique the
(Assessment Activity)
CGE5a, 5b, 5e
Teamwork
work of peers.
2 hours
1.4
ODV.01, OD1.03,
Knowledge
Use Fathom to retrieve
Using Fathom to
STV.01, ST1.04,
Teamwork
identified data sets and
Organize Data
STV.02, ST2.01,
Works Independently create a database for one(*Instructional Activity)
ST2.02, STV.03,
and two-variable
ST3.02, STV.04,
analyses.
4 hours
ST4.01, ST4.02,
ST4.03
CGE2b, 2c
1.5
ODV.01, OD1.03,
Knowledge
Use Fathom to retrieve,
Finding Data to Answer
STV.01, ST1.04,
Inquiry
organize, and analyse
Questions using Fathom
DMV.01, DM1.02,
Application
data from secondary
(Assessment Activity)
DMV.02, DM2.03
Communication
sources.
CGE2b, 2c, 3d, 4e
4 hours
(could include
assessment of
ST2.01, ST2.02,
ST3.02, ST4.01,
ST4.02)
1.6
ODV.01, OD1.02,
Knowledge
Investigate and critique
Looking Critically at
STV.01, ST1.04
the usefulness of
Websites
CGE1d, 3e, 5a, 5e
different sites; search for
2 hours
other useful data sites.
Page 4
 Mathematics of Data Management – University Preparation
Activity
Expectations
Assessment
Tasks
1.7
ODV.01, OD1.01,
Knowledge
Pose a significant
Culminating Activity for OD1.02, OD1.03,
Inquiry
problem, use the Internet
Unit 1
DMV.01, DM1.01,
Communication
to find data, organize and
DM1.02, DM1.03,
Application
analyse the data, and
5 hours
DMV.02 DM2.03
produce a poster; critique
CGE3b, 3e, 4e, 4f
the work of peers.
* Integration with Unit 2 is discussed within the activity description. Any additional time can be
allocated for remediation and consolidation of skills at the discretion of the teacher, depending on the
needs of students.
Unit 2: Data Analysis
Time: 23 hours
Ontario Catholic School Graduate Expectations: 2e, 3b, 3c, 3d, 5a
Unit Description
Students learn techniques for sampling data, including awareness of bias. They apply the common
techniques used for analysing one- and two-variable data and they learn to evaluate and critique the use
and misuse of statistics. Catholic students integrate their Catholic faith tradition as reflective and creative
thinkers making decisions in light of gospel values.
Unit Overview Chart
Cluster
Expectations
Assessment
Focus
STV.01, ST1.01, ST1.02,
Knowledge
Demonstrate an understanding of
1
ST1.03
Communication
standard techniques for collecting
CGE2e, 3b, 5a
Application
data and of different types of bias
STV.02, ST2.01, ST2.02,
Knowledge
Compute and interpret measures of
2
ST2.03
Communication
one-variable statistics using a
CGE3c, 3d
Application
variety of techniques.
STV.03, ST3.01, ST3.02,
Knowledge
Solve problems involving the
ST3.03
Inquiry
normal distribution.
3
CGE3c, 3d
Communication
Application
STV.04, ST4.01, ST4.02,
Knowledge
Describe the relation between two
4
ST4.03, ST4.04
Communication
variables by interpreting the
CGE3c, 3d
Application
correlation coefficient.
STV.05, ST5.01, ST5.02,
Knowledge
Evaluate the validity of statistics
ST5.03
Inquiry
drawn from a variety of sources.
5
CGE2e, 3b, 3c, 3d
Communication
Application
Unit 3: Counting and Probability
Time: 20 hours
Ontario Catholic School Graduate Expectations: 3b, 3c, 3d, 5b
Unit Description
Students develop skills for counting and determining probabilities using Venn diagrams, simulations,
counting principles, factorial notation, permutations, and combinations. They consider experimental and
theoretical probability, calculate expected values, and use the binomial distribution.
Page 5
 Mathematics of Data Management – University Preparation
Unit Overview Chart
Cluster
Expectations
CPV.01, CP1.01, CP1.02,
CP1.08
1
CGE 3b, 3c
2
3
4
5
6
CPV.01, CP1.03, CP1.04,
CP1.05, CP1.06, CP1.08
CGE 5b
CPV.01, CP1.07, CP1.08
CGE 3b, 3c
CPV.02, CP2.01, CP2.06,
OD2.02
CGE 3b, 3d
CPV.02, CP2.02, CP2.03,
CP2.04, CP2.06, CPV.03,
CP3.02
CGE 3c, 3d
CPV.02, CP2.05, CP2.06
CGE 3b
Assessment
Knowledge
Communication
Application
Knowledge
Inquiry
Communication
Application
Knowledge
Communication
Application
Knowledge
Application
Communication
Knowledge
Inquiry
Communication
Application
Knowledge
Inquiry
Communication
Application
Focus
Solve introductory counting
problems using Venn diagrams
together with the additive and
multiplicative counting principles.
Solve problems involving
permutations and combinations.
Connect Pascal’s Triangle with
binomial expansions.
Use counting techniques to solve
simple probability problems.
Determine expected values and
interpret them within the context
of an application.
Use the binomial distribution
model to determine probabilities.
Unit 4: Additional Tools for Data Management
Time: 20 hours
Ontario Catholic School Graduate Expectations: 3b, 3c, 3d, 5b
Unit Description
Students use matrices to organize and analyse data. Concepts and skills, understood and practised using
small data sets, can be applied to large data sets with the use of technology.
Unit Overview Chart
Cluster
Expectations
Assessment
Focus
ODV.02, OD2.01, OD2.02
Knowledge
Investigate situations that can be
CGE 3b, 3c
Application
modelled using diagrams (e.g., tree
1
Communication
diagrams, network diagrams,
cause-and-effect diagrams).
ODV.02, OD2.03
Application
Solve network problems using
2
CGE 3c, 3d, 5b
Inquiry
introductory graph theory.
ODV.03, OD3.01
Communication
Use matrices as a tool for
3
CGE 3b, 3c
Application
organizing data; develop the
related terminology and notation.
ODV.03, OD3.02
Knowledge
Develop proficiency with matrix
CGE 3b
Application
operations, such as addition, scalar
4
multiplication, and matrix
multiplication, with and without
the use of technology.
Page 6
 Mathematics of Data Management – University Preparation
Cluster
5
Expectations
ODV.03, OD3.03
CGE 3c, 3d, 5b
Assessment
Application
Inquiry
Focus
Apply matrix tools to solve
problems drawn from a variety of
applications.
Unit 5: Managing the Culminating Project
Time: 26 hours
Ontario Catholic School Graduate Expectations: 1d, 1i, 2a, 2b, 3c, 3b, 5a, 5g, 5e
Unit Description
Students prepare to successfully complete the culminating project outlined in the Integration of the
Techniques of Data Management strand. Students engage in activities in which they apply several of the
techniques/tools of the course to answer significant questions. Each activity could be viewed as a miniproject, providing the teacher with a vehicle for giving each student an opportunity to make a
presentation to the class and have it critiqued by other students. The student gains valuable experience
with these two expectations, which form part of the culminating project.
Unit Overview Chart
Activity
Expectations
Assessment
Tasks
5.1
DMV.01, DM1.01,
Application
Sequenced planning
Stages of the
DM1.02, DM1.03
Communication approach to the culminating
Culminating Project
CGE1i
project
8 hours
5.2
ODV.01, OD1.01, OD1.03, Knowledge
Analyse a set of data about
Income in Canadian
STV.01, ST1.04, STV.02,
Inquiry
family income in Canada
Families
ST2.01, ST2.02, ST2.03,
Communication and then pose and answer
STV.05, ST5.03
Application
questions about it.
4 hours
CGE3c
5.3
ODV.01, OD1.02, CPV.03, Knowledge
Analyse data related to the
AIDS in Canada
CP3.01, CP3.02, STV.01,
Inquiry
spread of AIDS in Canada
ST1.04, STV.04, ST4.01,
Communication over 20 years and construct
4 hours
ST4.02, ST4.04
Application
a simulation to compare and
CGE1d, 2a
contrast with the actual data.
5.4
CPV.02, CPV.03, CP2.01,
Inquiry
Examine selected games to
Dice Differences and
CP2.02, CP2.04, CP2.05,
Application
compare experimental and
the Non-Transitivity
CP2.06, CP3.01, CP3.02,
theoretical probability.
Paradox
CP3.03
3 hours
CGE5a, 3b
5.5
ODV.02, OD2.01, OD2.02, Application
Introduction to possible
Using Unit 4 Tools
OD2.03, ODV.03, OD3.01,
areas for further study by the
OD3.02, OD3.03
use of famous ideas
2 hours
CGE2b, 3b
5.6
DMV.01, DM1.03,
Communication Feedback, tools, and
Presentations and
DMV.02, DM2.01,
Knowledge
guidance in the preparation
Critiquing
DM2.02, DM2.03
of presentations
5 hours
CGE5e, 5g
Page 7
 Mathematics of Data Management – University Preparation
Teaching/Learning Strategies
To address the wide range of expectations in this course, a variety of teaching, learning, and assessment
strategies and tools need to be used. Teachers assume a variety of roles (including guide, facilitator,
consultant, and instructor) and employ a variety of strategies, including:
 a balance of whole-class, small group, mixed-ability structured group, and individual instruction
through student-centred and teacher-directed activities (group work should be carefully structured
along cooperative learning principles to be effective);
 the use of rich contextual problems which engage students and provide them with opportunities to
demonstrate learning and to appreciate the need for new skills;
 the prompting, supporting, and challenging of individual students, as well as the class as a whole;
 approaches that accommodate multiple learning styles (e.g., the provision of verbal and written
instructions, the inclusion of hands-on activities, etc.);
 the use of technological tools and software (e.g., graphing software, dynamic geometry software, the
Internet, spreadsheets, and multimedia) in activities, demonstrations, and investigations to facilitate
the exploration and understanding of mathematical concepts;
 the use of learning/performance tasks that are designed to link several expectations and give students
occasion to demonstrate their optimal levels of achievement through the demonstration of skill
acquisition, the communication of results, the ability to pose extending questions following an
inquiry, and the determination of a solution to unfamiliar problems;
 the use of accommodations, remediation, and/or extension activities;
 opportunities for students to practise and extend their skills and knowledge outside of the classroom.
Students themselves should play an active role in their own learning. To successfully complete the
requirements of this course, students are expected to:
 develop an increased responsibility for their own learning;
 be accountable for prerequisite skills;
 participate as active learners;
 engage in explorations using technology;
 apply individual and group learning skills;
 describe verbally, algebraically, and visually the mathematical patterns that emerge.
Assessment & Evaluation of Student Achievement
Assessment, as defined in the document Ontario Secondary Schools, Grades 9-12, Program and
Diploma Requirements, 1999, is “the process of gathering information from a variety of sources
(including assignments, demonstrations, projects, performances, and tests) that accurately reflects how
well students are achieving the curriculum expectations” (p. 31). Assessment tools should be designed to
allow students to demonstrate the full extent of their learning across the four categories of the
Achievement Chart. As teachers use a variety of assessment tools, it is necessary to ensure that a
consistent standard is maintained. Tools should be developed with the learning expectations of the course
as the criteria for this standard.
Students’ effective demonstration of communication skills is an essential component when evaluating
achievement. Students are required to display and convey their knowledge and understanding of
concepts, share their process of thought and inquiry, and justify their application of concepts in an
unfamiliar situation. In addition, their ability to communicate these skills is also assessed.
Teachers must continue to expand their understanding of Application skills to include non-routine
applications. This view requires a shift from the specific application of concepts (i.e., familiar situations),
to the general application of concepts (i.e., unfamiliar situations).
Page 8
 Mathematics of Data Management – University Preparation
Assessment strategies and tools must address a variety of teaching and learning styles in addition to the
criteria established by the learning expectations. Tests consisting only of questions that ask students to
perform algorithms and apply their knowledge do not necessarily offer an opportunity for students to
demonstrate Level 4 performance.
It is understood that students will meet course expectations at a variety of performance levels. An
effective and well-balanced assessment program provides students with several opportunities to
demonstrate growth and improvement over time, across all of the knowledge and skill categories.
Evaluation, as defined by Ontario Secondary Schools, Grades 9-12, Program and Diploma
Requirements, 1999, is “the process of judging the quality of a student’s work on the basis of established
achievement criteria, and assigning a value to represent that quality” (p. 31). Assessment is the collection
of information about student performance; evaluation is the determination of a quantitative value
describing the student’s overall level of achievement. Effective assessment, evaluation, and reporting
require the teacher to do more than average marks. Averaging is not comprehensive enough for accurate
reporting. As students can be expected to improve their performances over time, emphasis should be
placed on their most recent and most consistent level of achievement.
Seventy per cent of the grade will be based on assessments conducted throughout the course. Thirty per
cent of the grade will be based on a final evaluation which would include a combination of a formal
examination and a culminating performance task (student project presentation and critiques). It would be
reasonable to weight the culminating performance task higher than the final examination (e.g.,
culminating performance task twenty percent and the final examination ten percent).
Assessment Strategies
An effective assessment program includes a balance of diagnostic, formative, and summative assessment
instruments that incorporate the categories defined in the Achievement Chart for Mathematics. The
following are examples of strategies.
Knowledge/
Thinking/Inquiry/
Communication Application
Understanding
Problem Solving
final examinations




journals



observations



oral presentations


performance tasks




portfolios




quizzes

reports/assignments



student/teacher


conferences
unit tests




Assessment tools, such as observational checklists, performance criteria, rubrics, the Achievement Chart,
marking schemes, rating scales, peer evaluation, and self-evaluation, are used to assist in developing
objective and consistent evaluations of student achievement.
Page 9
 Mathematics of Data Management – University Preparation
Accommodations
Teachers refer to and contribute to Individual Education Plans (IEPs) of students and consider their
particular learning characteristics to make necessary accommodations. Teachers work in consultation
with resource teachers, ESL/ELD teachers, where available, and parents or guardians to determine
appropriate accommodations to support effective student learning and assessment.
Specific Accommodations
 Provide a list of terms (possibly simplified) before an activity begins.
 Modify handouts in terms of the terminology and content used, as well as the size and typeface of the
selected font. Allow plenty of space for written responses.
 Allow assignments to be completed in alternate formats or using longer timelines.
 Keep manipulatives, grid paper, formula sheets, and other aids available for needs that arise.
 Provide students with oral pre-planning of activities.
 Give more time to complete written work (copying from the board proofreading).
 Have students produce work using a word-processing package on a computer.
 Allow students to read pertinent text into a recording device, such as an audio tape recorder.
 Give several short assignments rather than one long one.
 Use oral presentation.
 Provide overhead copies before the class begins.
 Describe using diagrams, charts, and graphs. Reinforce verbally.
 Have interesting, relevant books and articles available that are at the appropriate reading level.
 Have all responses given in a written format.
 Do not ask the student to respond to questions without forewarning.
Alternative Assessment Techniques
 Use oral tests; give open-book tests or use of notes; give tests that elicit short answers and multiple
choice, true/false, matching tests; use short quizzes instead of major tests.
 Assign fewer questions, especially in research projects if the student is unable to indicate that he/she
comprehends and has mastered task.
 Tape tests. Student listens and/or responds on tape.
 Extend time on tests.
 Give tasks that allow for a variety of responses, visual, oral, etc.
 Have ESL students work in pairs, with peer tutors, with classmates that have the same linguistic
background, or with cooperative supportive groups, where they are more likely to improve their use
of English. Brainstorm in groups using the students’ first language if their usage of English is
limited.
 Use peer conferencing to reinforce instructions or information.
 Provide reference notes, outlines of critical information, models of charts, timelines, or diagrams.
 Use visuals to illustrate definitions for the students’ dictionary of terms.
 Pair written instructions with verbal instructions. Provide visual or auditory cues.
 Simplify instructions. Highlight key words or phrases.
 Reinforce main ideas by using the think/pair/share peer-assessment strategy.
 Provide opportunities for students to practise oral presentation skills.
Page 10
 Mathematics of Data Management – University Preparation
Resources
Units in this Course Profile make reference to the use of specific texts, magazines, films, videos, and
websites. The teachers need to consult their board policies regarding use of any copyrighted materials.
Before reproducing materials for student use from printed publications, teachers need to ensure that their
board has a Cancopy licence and that this licence covers the resources they wish to use. Before screening
videos/films with their students, teachers need to ensure that their board/school has obtained the
appropriate public performance videocassette licence from an authorized distributor, e.g., Audio Cine
Films Inc. The teachers are reminded that much of the material on the Internet is protected by copyright.
The copyright is usually owned by the person or organization that created the work. Reproduction of any
work or substantial part of any work from the Internet is not allowed without the permission of the
owner.
The URLs for the websites were verified by the writers prior to publication. Given the frequency with
which these designations change, teachers should verify the websites prior to assigning them for student
use.
Fathom, TI-Interactive, graphing calculators (e.g., TI-83+)
Spreadsheet software (e.g., Quattro Pro, Excel)
Internet access
Statistics Canada – www.statcan.ca or http://estat.statcan.ca
Environment Canada – www.ec.gc.ca
School library/resource centre for guides to help students in preparing essays, bibliographies, etc.
Teachers in other departments can also be used as resources.
OSS Considerations
The following resources support many of the Ontario Secondary School policies, as well as the Ontario
Catholic School Graduate Expectations.
Ministry of Education Policy and Reference Documents
Choices Into Action: Guidance and Career Education Program Policy, 1999.
Cooperative Education: Policies and Procedures for Ontario Secondary Schools, 2000.
Individual Education Plans: Standards for Development, Program Planning, and Implementation, 2000.
The Ontario Curriculum, Mathematics, Grades 9-10, 1999.
The Ontario Curriculum, Mathematics, Grades 11-12, 2000.
Ontario Schools Code of Conduct.
Ontario Secondary Schools, Grades 9-12, Program and Diploma Requirements, 1999.
Program Planning and Assessment, Grades 9-12, 2000.
Violence-Free Schools Policy.
The Ministry of Education has published several resource documents, brochures, and policy/program
memoranda in support of its OSS policies, available online (www.edu.gov.on.ca).
Page 11
 Mathematics of Data Management – University Preparation
Coded Expectations, Mathematics of Data Management, Grade 12,
University Preparation, MDM4U
Organization of Data for Analysis
Overall Expectations
ODV.01 · organize data to facilitate manipulation and retrieval;
ODV.02 · solve problems involving complex relationships, with the aid of diagrams;
ODV.03 · model situations and solve problems involving large amounts of information, using matrices.
Specific Expectations
Organizing Data
OD1.01 – locate data to answer questions of significance or personal interest, by searching wellorganized databases;
OD1.02 – use the Internet effectively as a source for databases;
OD1.03 – create database or spreadsheet templates that facilitate the manipulation and retrieval of data
from large bodies of information that have a variety of characteristics (e.g., a compact disc collection
classified by artist, by date, by type of music).
Using Diagrams to Solve Problems
OD2.01 – represent simple iterative processes (e.g., the water cycle, a person’s daily routine, the creation
of a fractal design), using diagrams that involve branches and loops;
OD2.02 – represent complex tasks (e.g., searching a list by using algorithms; classifying organisms;
calculating dependent or independent outcomes in probability) or issues (e.g., the origin of global
warming), using diagrams (e.g., tree diagrams, network diagrams, cause-and-effect diagrams, time
lines);
OD2.03 – solve network problems (e.g., scheduling problems, optimum-path problems, critical-path
problems), using introductory graph theory.
Using Matrices to Model and Solve Problems
OD3.01 – represent numerical data, using matrices, and demonstrate an understanding of terminology
and notation related to matrices;
OD3.02 – demonstrate proficiency in matrix operations, including addition, scalar multiplication, matrix
multiplication, the calculation of row sums, and the calculation of column sums, as necessary to
solve problems, with and without the aid of technology;
OD3.03 – solve problems drawn from a variety of applications (e.g., inventory control, production costs,
codes), using matrix methods.
Counting and Probability
Overall Expectations
CPV.01 · solve counting problems and clearly communicate the results;
CPV.02 · determine and interpret theoretical probabilities, using combinatorial techniques;
CPV.03 · design and carry out simulations to estimate probabilities.
Page 12
 Mathematics of Data Management – University Preparation
Specific Expectations
Solving Counting Problems
CP1.01 – use Venn diagrams as a tool for organizing information in counting problems;
CP1.02 – solve introductory counting problems involving the additive and multiplicative counting
principles;
CP1.03 – express the answers to permutation and combination problems, using standard combinatorial
symbols, [e.g.,  nr  , P(n, r)];
 
CP1.04 – evaluate expressions involving factorial notation, using appropriate methods (e.g., evaluating
mentally, by hand, by using a calculator);
CP1.05 – solve problems, using techniques for counting permutations where some objects may be alike;
CP1.06 – solve problems, using techniques for counting combinations;
CP1.07 – identify patterns in Pascal’s triangle and relate the terms of Pascal’s triangle to values of  nr  ,
 
to the expansion of a binomial, and to the solution of related problems (Sample problem: A girl’s
school is 5 blocks west and 3 blocks south of her home. Assuming that she leaves home and walks
only west or south, how many different routes can she take to school?);
CP1.08 – communicate clearly, coherently, and precisely the solutions to counting problems.
Determining and Interpreting Theoretical Probabilities
CP2.01 – solve probability problems involving combinations of simple events, using counting techniques
[i.e., P(A or B), P(A and B), and P(~A)];
CP2.02 – identify examples of discrete random variables (e.g., the sums that are possible when two dice
are rolled);
CP2.03 – construct a discrete probability distribution function by calculating the probabilities of a
discrete random variable;
CP2.04 – calculate expected values and interpret them within applications (e.g., lottery prizes, tests of
the fairness of games, estimates of wildlife populations) as averages over a large number of trials;
CP2.05 – determine probabilities, using the binomial distribution (Sample problem: A light-bulb
manufacturer estimates that 0.5% of the bulbs manufactured are defective. If a client places an order
for 100 bulbs, what is the probability that at least one bulb is defective?);
CP2.06 – interpret probability statements, including statements about odds, from a variety of sources.
Simulating and Predicting
CP3.01 – identify the advantages of using simulations in contexts;
CP3.02 – design and carry out simulations to estimate probabilities in situations for which the
calculation of the theoretical probabilities is difficult or impossible (Sample problem: A set of 6
baseball cards can be collected from cereal boxes. If the different cards are evenly distributed
throughout the boxes, carry out a simulation to determine the probability of collecting one complete
set in a purchase of 14 boxes);
CP3.03 – assess the validity of some simulation results by comparing them with the theoretical
probabilities, using the probability concepts developed in the course (Sample problem: A light-bulb
manufacturer estimates that 0.5% of the bulbs manufactured are defective. Carry out a simulation to
estimate the probability that at least one bulb is defective in an order of 100 bulbs).
Page 13
 Mathematics of Data Management – University Preparation
Statistics
Overall Expectations
STV.01 · demonstrate an understanding of standard techniques for collecting data;
STV.02 · analyse data involving one variable, using a variety of techniques;
STV.03 · solve problems involving the normal distribution;
STV.04 · describe the relationship between two variables by interpreting the correlation coefficient;
STV.05 · evaluate the validity of statistics drawn from a variety of sources.
Specific Expectations
Collecting Data
ST1.01 – demonstrate an understanding of the purpose and the use of a variety of sampling techniques
(e.g., a simple random sample, a systematic sample, a stratified sample);
ST1.02 – describe different types of bias that may arise in surveys (e.g., response bias, measurement bias,
non-response bias, sampling bias);
ST1.03 – illustrate sampling bias and variability by comparing the characteristics of a known population
with the characteristics of samples taken repeatedly from that population, using different sampling
techniques;
ST1.04 – organize and summarize data from secondary sources (e.g., the Internet, computer databases),
using technology (e.g., spreadsheets, graphing calculators).
Analysing Data Involving One Variable
ST2.01 – compute, using technology, measures of one-variable statistics (i.e., the mean, median, mode,
range, interquartile range, variance, and standard deviation), and demonstrate an understanding of the
appropriate use of each measure;
ST2.02 – interpret one-variable statistics to describe characteristics of a data set;
ST2.03 – describe the position of individual observations within a data set, using z-scores and
percentiles.
Solving Problems Involving the Normal Distribution
ST3.01 – identify situations that give rise to common distributions (e.g., bimodal, U-shaped, exponential,
skewed, normal);
ST3.02 – demonstrate an understanding of the properties of the normal distribution (e.g., the mean,
median, and mode are equal; the curve is symmetric about the mean; 68% of the population are
within one standard deviation of the mean) and use these properties to solve problems;
ST3.03 – make probability statements about normal distributions, on the basis of information drawn from
a variety of applications.
Describing the Relationship Between Two Variables
ST4.01 – define the correlation coefficient as a measure of the fit of a scatter graph to a linear model;
ST4.02 – calculate the correlation coefficient for a set of data, using graphing calculators or statistical
software;
ST4.03 – demonstrate an understanding of the distinction between cause-effect relationships and the
mathematical correlation between variables;
ST4.04 – describe possible misuses of regression (e.g., use with too small a sample, use without
considering the effect of outliers, inappropriate extrapolation).
Page 14
 Mathematics of Data Management – University Preparation
Evaluating Validity
ST5.01 – explain examples of the use and misuse of statistics in the media;
ST5.02 – assess the validity of conclusions made on the basis of statistical studies, by analyzing possible
sources of bias in the studies (e.g., sampling bias) and by calculating and interpreting additional
statistics, where possible (e.g., measures of central tendency, the standard deviation);
ST5.03 – explain the meaning and the use in the media of indices based on surveys (e.g., the consumer
price index, the cost of living index).
Integration of the Techniques of Data Management
Overall Expectations
DMV.01 · carry out a culminating project on a topic or issue of significance that requires the integration
and application of the expectations of the course;
DMV.02 · present a project to an audience and critique the projects of others.
Specific Expectations
Carrying Out a Culminating Project
DM1.01 – pose a significant problem whose solution would require the organization and analysis of a
large amount of data;
DM1.02 – select and apply the tools of the course (e.g., methods for organizing data, methods for
calculating and interpreting measures of probability and statistics, methods for data collection) to
design and carry out a study of the problem;
DM1.03 – compile a clear, well-organized, and fully justified report of the investigation and its findings.
Presenting and Critiquing Projects
DM2.01 – create a summary of a project to present within a restricted length of time, using
communications technology effectively;
DM2.02 – answer questions about a project, fully justifying mathematical reasoning;
DM2.03 – critique the mathematical work of others in a constructive fashion.
Page 15
 Mathematics of Data Management – University Preparation
Ontario Catholic School Graduate Expectations
The graduate is expected to be:
A Discerning Believer Formed in the Catholic Faith Community who
CGE1a -illustrates a basic understanding of the saving story of our Christian faith;
CGE1b -participates in the sacramental life of the church and demonstrates an understanding of the
centrality of the Eucharist to our Catholic story;
CGE1c -actively reflects on God’s Word as communicated through the Hebrew and Christian
scriptures;
CGE1d -develops attitudes and values founded on Catholic social teaching and acts to promote social
responsibility, human solidarity and the common good;
CGE1e -speaks the language of life... “recognizing that life is an unearned gift and that a person
entrusted with life does not own it but that one is called to protect and cherish it.” (Witnesses
to Faith)
CGE1f -seeks intimacy with God and celebrates communion with God, others and creation through
prayer and worship;
CGE1g -understands that one’s purpose or call in life comes from God and strives to discern and live
out this call throughout life’s journey;
CGE1h -respects the faith traditions, world religions and the life-journeys of all people of good will;
CGE1i -integrates faith with life;
CGE1j -recognizes that “sin, human weakness, conflict and forgiveness are part of the human
journey” and that the cross, the ultimate sign of forgiveness is at the heart of redemption.
(Witnesses to Faith)
An Effective Communicator who
CGE2a -listens actively and critically to understand and learn in light of gospel values;
CGE2b -reads, understands and uses written materials effectively;
CGE2c -presents information and ideas clearly and honestly and with sensitivity to others;
CGE2d -writes and speaks fluently one or both of Canada’s official languages;
CGE2e -uses and integrates the Catholic faith tradition, in the critical analysis of the arts, media,
technology and information systems to enhance the quality of life.
A Reflective and Creative Thinker who
CGE3a -recognizes there is more grace in our world than sin and that hope is essential in facing all
challenges;
CGE3b -creates, adapts, evaluates new ideas in light of the common good;
CGE3c -thinks reflectively and creatively to evaluate situations and solve problems;
CGE3d -makes decisions in light of gospel values with an informed moral conscience;
CGE3e -adopts a holistic approach to life by integrating learning from various subject areas and
experience;
CGE3f -examines, evaluates and applies knowledge of interdependent systems (physical, political,
ethical, socio-economic and ecological) for the development of a just and compassionate
society.
Page 16
 Mathematics of Data Management – University Preparation
A Self-Directed, Responsible, Life Long Learner who
CGE4a -demonstrates a confident and positive sense of self and respect for the dignity and welfare of
others;
CGE4b -demonstrates flexibility and adaptability;
CGE4c -takes initiative and demonstrates Christian leadership;
CGE4d -responds to, manages and constructively influences change in a discerning manner;
CGE4e -sets appropriate goals and priorities in school, work and personal life;
CGE4f -applies effective communication, decision-making, problem-solving, time and resource
management skills;
CGE4g -examines and reflects on one’s personal values, abilities and aspirations influencing life’s
choices and opportunities;
CGE4h -participates in leisure and fitness activities for a balanced and healthy lifestyle.
A Collaborative Contributor who
CGE5a -works effectively as an interdependent team member;
CGE5b -thinks critically about the meaning and purpose of work;
CGE5c -develops one’s God-given potential and makes a meaningful contribution to society;
CGE5d -finds meaning, dignity, fulfillment and vocation in work which contributes to the common
good;
CGE5e -respects the rights, responsibilities and contributions of self and others;
CGE5f -exercises Christian leadership in the achievement of individual and group goals;
CGE5g -achieves excellence, originality, and integrity in one’s own work and supports these qualities
in the work of others;
CGE5h -applies skills for employability, self-employment and entrepreneurship relative to Christian
vocation.
A Caring Family Member who
CGE6a -relates to family members in a loving, compassionate and respectful manner;
CGE6b -recognizes human intimacy and sexuality as God given gifts, to be used as the creator
intended;
CGE6c -values and honours the important role of the family in society;
CGE6d -values and nurtures opportunities for family prayer;
CGE6e -ministers to the family, school, parish, and wider community through service.
A Responsible Citizen who
CGE7a -acts morally and legally as a person formed in Catholic traditions;
CGE7b -accepts accountability for one’s own actions;
CGE7c -seeks and grants forgiveness;
CGE7d -promotes the sacredness of life;
CGE7e -witnesses Catholic social teaching by promoting equality, democracy, and solidarity for a
just, peaceful and compassionate society;
CGE7f -respects and affirms the diversity and interdependence of the world’s peoples and cultures;
CGE7g -respects and understands the history, cultural heritage and pluralism of today’s contemporary
society;
CGE7h -exercises the rights and responsibilities of Canadian citizenship;
CGE7i -respects the environment and uses resources wisely;
CGE7j -contributes to the common good.
Page 17
 Mathematics of Data Management – University Preparation
Unit 1: Posing Questions With Data
Time: 21 hours
Unit Description
Students learn to find, retrieve, and organize credible data. They learn to pose significant questions
through the use of journals and critique the work of others. Some activities are grouped to teach the
expectations in an instructional activity followed by an assessment activity.
Using Fathom, students locate and retrieve large data sets from a variety of Internet sites, including
Statistics Canada (E-STAT). Students answer questions using the data sets and consider and explore
other factors that could influence the collection of data. Students use the features of Fathom to analyse
one- and two-variable data; analyses include cause-and-effect and regression. Students present their
findings in small-group settings and critique the data analyses of others clearly, honestly, and with
sensitivity. Students complete the unit by posing a problem, finding and analysing data, presenting their
work on a poster, and critiquing the work of others.
Throughout the unit, students keep a journal in which they reflect on their responses to questions, posed
in a way that demonstrates their respect for the rights, responsibilities, and contributions of self and
others. Through class discussions, students in Catholic schools develop attitudes and values founded on
Catholic social teaching.
The activities require the use of computers with Internet access. It is recommended that this course be
scheduled in a computer lab. If this is not possible, computers should be reserved for a minimum of 20
hours. The 20 hours can be reserved as a solid block with Unit 1 taught as outlined. It is also possible to
block smaller amounts of time and teach the data analysis tools of Unit 2 between the lab days.
Following this plan, students would be expected to complete their data analysis using the additional
tools.
Note: An independent assignment in Activity1.6, requiring that students design a checklist to identify the
strengths, shortcomings, validity, etc. websites, should be assigned at the beginning of the course. The
assignment can be completed individually or in pairs.
Journal Organization
A journal, in the form of a binder or folder, is used as an organizer for the course and gives teachers an
opportunity to provide formative feedback. The following suggested sections are referenced:
 Ethical Implications of Data – reflections about social issues and considerations of possible motives
(and bias) for collecting data;
 Data Sites – record of all data sites and a summary of the credibility of the data; some type of
classification should be used to organize data sites;
 Posing Questions/Problems – used to consider what makes a good problem/question; teachers should
use this section to provide formative assessment on posing a problem;
 Critiquing – reactions to critiques;
 Deadlines – record of deadlines for assignments and checkpoints for final project;
 Final Project Brainstorm – any sites, data articles that interest students. Students may want to use the
articles for their final projects.
Unit 1 – Page 1
 Mathematics of Data Management – University Preparation
Resources
Data
Statistics Canada – www.statcan.ca or http://estat.statcan.ca
Environment Canada – www.ec.gc.ca
Transport Canada – www.tc.gc.ca
Key Curriculum Press – www.keypress.com/Fathom
Data and Story Library – http://lib.stat.cmu/DASL
Exploring Data – Introductory Statistics – http://exploringdata.cqu.edu.au
Quantitative Environmental Learning Project – www.seattlecentral.org/qelp/
Internet Site Credibility – http://cybrary.uwinnipeg.ca/
Activity 1.1: Posing Questions and Reading Graphs
Time: 1 hour
Description
This activity is a launching pad for the focus of Unit 1 – posing questions and finding data. In groups,
students create questions concerning a given set of graphs and then answer questions designed by another
group. Students also begin an assignment from Activity 1.6 that will be used at a later date. Students have
an opportunity to discuss democracy and solidarity for a just, peaceful, and compassionate society and
students in Catholic schools will relate them to Catholic teachings.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE5a - works effectively as an interdependent team member;
CGE7e - witness Catholic social teaching by promoting equality, democracy, and solidarity for a just,
peaceful, and compassionate society.
Strand(s): Integration of the Techniques of Data Management
Overall Expectations
DMV.02 - present a project to an audience and critique the projects of others.
Specific Expectations
DM2.02 - answer questions about a project, fully justifying mathematical reasoning.
Prior Knowledge & Skills
 Read and interpret a variety of different graphs (bar and broken-line graphs/dot and scatter plots).
Planning Notes
 Photocopy the Student Activity handout.
 Assign Activity 1.6 assignment – How to Determine if a Website is Credible.
Teaching/Learning Strategies
For the first 10 minutes of class, the teacher and students brainstorm and review graphs, including
independent variable, dependent variable, trends in data, the different types of trends or models students
have previously seen, and reading points from graph. Place students in groups of three and distribute the
handout. Students create questions that can be answered from the graphs. Groups then switch papers and
answer the questions; students should critically look at the questions to see if they truly can be answered
from the graph. For the remaining class time, groups present their questions and solutions. As students
Unit 1 – Page 2
 Mathematics of Data Management – University Preparation
present their work in groups, the class gives constructive feedback as to whether they agree or disagree
with the groups’ findings. The depth and richness of the questions is discussed (using waste graph and
population graph to determine and compare waste per capita in Canada for 1996).
Journal Reflection: Students write a reflection of their thoughts and values for each graph, which
become important when students evaluate websites in later activities. The ethical implication of the data
can provide insight into the motive of posting the website and determining any bias in the data or its
findings.
Student Activity
On a separate sheet of paper, create questions that could be answered by the given series of graphs. Your
questions should be both factual and theoretical. Design questions that may be answered using one graph
and questions that can be answered using two related graphs. Switch your completed question sheet with
another group and answer the questions they have designed.
Chicago Murders in January and July (www.icpsr.umich.edu/nacjd/SDA/chd95.htm)
This graph depicts the relationship between the victim and the offender (who killed them). Points that
appear darker than the rest represent two points.
This graph depicts the sex of the victim and the sex of the offender.
Unit 1 – Page 3
 Mathematics of Data Management – University Preparation
Amount of Waste Disposed by Province (http://estat.statcan.ca/cgi-win/CNSMCGI.EXE)
Population of the Provinces in 1996
(http://estat.statcan.ca/cgi-win/CNSMCGI.EXE?Lang=E&DBSelect=SD1ALL)
Canadian Youth Property Crimes (theft, break-and-enter, fraud)
http://estat.statcan.ca/cgi-win/CNSMCGI.EXE
Youths Charged with Property Crimes
(theft, break and enter, fraud)
Note:
Top line: Property crimes
2nd line: Theft, over and under
3rd line: Breaking and entering
Unit 1 – Page 4
4th line: Theft, motor vehicle
5th line: Have stolen goods
6th line: Frauds
 Mathematics of Data Management – University Preparation
Air Pollution (http://lib.stat.cmu.edu/DASL/Datafiles/AirPollution.html)
SO2: Sulphur dioxide content of air in micrograms per cubic metre.
Temp: Average annual temperature in degrees Fahrenheit.
Journal Reflection: In Ethical Implications of Data, reflect on the usefulness and ethical implications of
this data. Could this data be used to improve society or the environment and by whom (the Church?
schools?) How could it be used constructively and could it also be misused? In Posing Problems, select
two questions from the class discussion and discuss the strengths and weaknesses of each question.
Assessment & Evaluation of Student Achievement
 Focus on formative assessment of student ideas/work during activity.
 Collect journals – or discuss journal entries about strengths and weaknesses of questions.
Resources
Background information on the Young Offenders Act and violent crimes can be found at:
– http://estat.statcan.ca/content/English/articles/cyb/cyb-just.htm
– http://www.statcan.ca/Daily/English/980127/d980127.htm#ART1
Activity 1.2: Exploring and Retrieving Data Using E-STAT
Time: 3 hours
Description
This activity is an introduction to E-STAT. Students navigate through E-STAT and learn the tools of ESTAT by investigating data concerning the number of Canadians who went to a casino; the number of
trees harvested and replenished by each province; and the production of beer and impaired driving
charges. Students copy data and graphs created by E-STAT into a word-processing document.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE2b - reads, understands, and uses written material effectively;
CGE3c - thinks reflectively and creatively to evaluate situations and solve problems;
CGE3d - decisions in light of gospel values with an informed moral conscience;
CGE7i - respects the environment and uses resources wisely.
Strand(s): Organization of Data for Analysis
Overall Expectations
ODV.01 - organize data to facilitate manipulation and retrieval.
Specific Expectations
OD1.01 - locate data to answer questions of significance or personal interest, by searching wellorganized databases;
OD1.02 - use the Internet effectively as a source for databases.
Unit 1 – Page 5
 Mathematics of Data Management – University Preparation
Prior Knowledge & Skills
 Read and interpret information from line, bar, and scatter graphs.
 Calculate slope.
Planning Notes
 Ensure that your school has E-STAT by going to www.statcan.ca >> English >> Learning Resources
>> ESTAT >> MEMBERS. If your school name is not on the list, contact board personnel; all school
boards in Ontario have access to ESTAT. For future information, data in E-STAT is free. If you are
asked to pay for data, you are no longer in E-STAT.
 Try E-STAT lesson in Part B Method 1 of Student Activity, (comparing trees harvested and
replenished in Canada) and photocopy it for students.
 Part A will take one hour; Part B, Method 1 will take one hour; Part B, Method 2 will take a half
hour. Method 1 can be shortened or lengthened depending on the number of years investigated.
 If computers are limited students could work in pairs and switch for each part.
 Use a multimedia projector to introduce Statistics Canada to the class and to solve common problems
or questions during the activity.
 In both oral and written discussions, students can discuss the moral and ethical implications of
gambling, respect for all human life and the earth, the ability to make free decisions, and the
consequences of those decisions in light of the gospel values
Teaching/Learning Strategies
Students keep a glossary of terms. As a possible assignment, italicized words should be defined in a
glossary for later reference when students are searching for their own data. After students finish writing
their summary for Part A, the class discusses the findings. The graphs do not necessarily support the
statement that casinos are successful in Canada; to answer that question, look at their revenues. Part B
demonstrates the different types of data in E-STAT. Method 1 is a ready-made lesson demonstrating the
creation of scatter plots with census and other types of data. The lesson can be printed and photocopied
for students to follow. Method 2 demonstrates creating scatter plots using CANSIM time series data.
There is not a strong relationship between beer production and impaired driving. It is important that
students understand that not all data leads to a relationship.
Journal Reflection: Students summarize their learning of the Statistics Canada website. They include
data they found interesting and reflect on the moral and ethical implications of the data (gambling,
forestry issues, drinking and driving). They create questions concerning the data that they feel are good
questions, which can done as a homework assignment. Provide feedback for questions.
Student Activity
Part A: Are Casinos Successful in Canada?
This activity is an introduction to the Statistics Canada website. You learn methods of data retrieval and
create line and bar graphs using E-STAT. Graphs are copied into a word-processing document and
summaries are written. You describe the set-up of Statistics Canada and how to access data in the Data
Sites section of your journal. You will be assessed at a later date on your ability to navigate through ESTAT to find, analyse, and display data.
Using E-STAT
1. Go to website www.statcan.ca >> English >> Learning Resources >> E-STAT.
2. Read the Welcome Page and the Licence Agreement, then Accept.
3. If the page looks jumbled, Refresh (on toolbar).
4. You are looking at the Table of Contents of Statistics Canada’s E-STAT data. Across the top of the
Table of Contents are Overview, Articles, and Data. For each topic, look at the Overview of the
topic, or Articles about the topic, or Data related to the topic.
Unit 1 – Page 6
 Mathematics of Data Management – University Preparation
Selecting a Topic of Interest in ESTAT
5. Select Service Industries for an Overview of a particular Service Industry topic. Below the graph,
statcan tells you where you can access the data. Also below the graph are Articles and Data.
6. Select Articles. This provides a list of articles that relate to different Service Industry topics. Note:
this list is updated regularly as new articles become available.
7. Select Data (at bottom of page).
Getting Data for a Topic of Interest
8. Here is a list of topics that are related to the Service Industry. We will look at how many Canadians
have gone to a casino over the years.
9. Select Accommodation service industry for a list of data tables related to the Accommodation
Service Industry. There are active tables and terminated tables. While terminated tables may still be
used, they are not updated. The data on active tables are updated on a regular basis (monthly,
annually).
10. Select the code 426-0006. From the Subset Selection:
 Select Canada from 1st list.
 Select Went to a casino from the 2nd list (use the scroll bar).
 From the years, select the earliest year and the most recent year (click on arrows).
 Continue
11. Generally, we use the Time series (Option 2) rather than Table (Option 1) option.
12. Select Time Series >> Continue. Note: We will add more series later.
Graphing the Data of Interest
13. Output Format Selection lets you choose how to view the data. Select Graphic: line chart, which is
best for a relationship over along period of time (but it is sometimes useful to display the data as a
bar chart, table of values, etc.). Select Go to generate the graph.
14. Select Change Titles and change the title to “The Number of Canadians who have gone to a Casino”.
Importing the Graph into a Word-Processing Document
15. Right click on the graph. Select copy. Open a word-processing document and paste. You may have
to adjust the graph (e.g., in Microsoft Word double click and in Layout select square). Write a
summary about how this graph could help you answer the question, “Are casinos successful in
Canada”. Save the document, but do not close it.
Graphing More Than One Data Set
16. Return to your Internet Browser. To compare the number of people who went to a casino who live in
Ontario to those who live in Quebec and British Columbia, use the back button until you are at
Subset selection again.
 While holding down the Control key (or the Command key on a Macintosh computer), select
Quebec, Ontario, and British Columbia.
 Select Went to a casino.
 Select earliest year and most recent year.
 Continue >> Time series >> Continue.
17. For Output Format Selection, select Graphic: vertical bar chart. Select Go. Change the title to a
more appropriate one. Copy the graph into the word-processing document.
18. Go back to E-STAT and Output Format Selection (use Back Button). Select Line chart. Copy the
resulting graph into your document. In your document, discuss which graph tells you more about the
casinos in Canada.
Unit 1 – Page 7
 Mathematics of Data Management – University Preparation
Viewing the Data as a Table of Values
19. You may notice that there are some data missing from the graphs. Go back to the Output Format
Selection. Select HTML Table, time as rows. At the bottom of the table is an explanation of why
some data are missing; they are labelled secure or confidential. Copy it into your document by
holding control and selecting the cells in the table. Then right click and paste into your word
processing document.
20. Write a final summary about the success of casinos in Canada using and referring to the data and
graphs. Print the document.
Journal Reflection: In Ethical Implications of Data, reflect on the moral and ethical implications of
gambling. In Posing Questions/Problems, create a question related to gambling that could be investigated
by analysing data.
Part B: Analysing Relationships Using Scatter Plots
Before importing data into a statistical program, it is important to see if there is a relationship between
two variables – is it worthwhile data? E-STAT has two methods for creating scatter graphs, depending on
the data you are using.
For CANSIM time series data (which shows trends over time), the numeric values of two characteristics
for each period of time become the x and y coordinates respectively for a single point on the scatter plot
(i.e., the two characteristics being graphed are matched for each time period and plotted as a single point
on the graph). The number of points on the scatter plot is exactly the number of time periods observed.
For Census and other geographic oriented data, the numeric values of two different characteristics for
each geographic area become the x and y coordinates respectively for one point on the scatter plot (i.e.,
the two characteristics being graphed are matched for each area and plotted as a single point). The
number of points on the scatter plot is exactly the number of geographic areas for which we have data.
Method 1: Census and Other Data
In E-STAT, there is a ready-made lesson to demonstrate the creation of scatter graphs for examining the
relationship between the volume of tree harvesting and tree planting for provinces in Canada.
To access this lesson:
 E-STAT >> For Teachers >> Lesson Plans >> Mathematics: Analysing Relationships Using
Scatter Graphs
In a word-processing document, students answer the questions in the activity.
Journal Reflection: In Ethical Implications of Data, reflect on the moral and ethical implications of
forestry. How does this data relate to your reflections? In Posing Questions/Problems, create a question
related to forestry that could be investigated by analysing data.
Method 2: CANSIM Data
Investigate if there is a relationship between beer production and impaired driving in Canada.
Getting the Data for Beer Production
 E-STAT >> Data >> Manufacturing >> Food Industries >> Production, bottling or stocks of
beverages, monthly (303-0019)
 Select: Canada for Geography, Beer Production (Litres) for Type of beverage.
 Select earliest to latest year (notice data are monthly).
 Continue >> Time series.
Adding Other Variables from Different Topics: Getting the Data for Impaired Driving
 Select Add more series; E-STAT has several methods to find other data sets of interest. If you
choose Subject, the initial E-STAT list of topics is displayed. The Keyword search is useful for
looking for a specific topic.
 Select Keyword.
 Under 3- Search for: type “impaired driving.” Press Enter.
 Select Table 255-0002 Actual traffic offences reported, by type of offence, Canada, provinces
and territories, annual (notice data are annual).
Unit 1 – Page 8
 Mathematics of Data Management – University Preparation


Select Canada for Geography.
There are many options for Type of traffic offence. To see the options better select View checklist
and footnotes. The format is easy to view and to select options click on box. Select Impaired
operation of motor vehicle or over 80 milligrams (section 237 Criminal Code). You must then
select Return to pick list at the bottom for your option to be selected. Do not press Back.
 Select Number of reported offences (actual) for Unit of measure.
 Select earliest and latest date >> Continue >> Time Series.
 You can see both of your selections. You could add more series. Notice the WARNING: Series do
not have the same Frequency (beer is monthly, impaired is annually). Select Continue.
Viewing Data Graphically: The Line Chart and the Scatter Graph.
 View data as a Graphic: line chart. Select Go. An Error message is displayed - the frequency of the
data is a problem since it is not the same. Press Back. This time under The frequency of the output
data will be converted to annual (sum). Now select Go. The graph is displayed. Press Back.
 View data as a Graphic: scatter graph (min. 2 series). Press Go. A scatter plot is produced
comparing the number of impaired drivers and the production of beer for the period that E-STAT
contains data. Notice E-STAT automatically puts a line of best fit through the data. When you
analyse the data, a line may not always be the best model to use. The scatter plot is more useful as a
check to see if there seems to be a clear relationship between the data variables and if this
relationship can be modelled by a line.
 Copy this graph into a word-processing document and discuss: What might this data imply? What
information may also be helpful to investigate? Is there a relationship? Would you investigate this
data further?
 What factors might explain the overall trend in impaired driving charges?
 What factors might explain the overall trend in beer production?
Journal Reflection: In Ethical Implications of Data, reflect on moral and ethical implications of drinking
and driving. How does this data relate to your reflections? In Posing Questions/Problem, create a
question related to beer production and drinking and driving that can be investigated by analysing data.
Assessment & Evaluation of Student Achievement
This is an instructional activity. Focus on formative assessment of knowledge and skills. Activity 1.3 is
the assessment activity that complements this activity. The questions created in the journals can be
formatively assessed by the teacher or peers could critique each others’ questions and decide if the
questions are too open or too narrow. Critiques should be written in the journal alongside questions.
Teachers can provide feedback for the critiques – are the critiques constructive and thorough? The
learning strategies of working independently and teamwork could be assessed.
Accommodations
This activity has a high density of words and instructions. Students who have difficulty with reading
instructions should check off the instructions they have accomplished so they are not lost when they
return to the instructions. The use of a highlighter would help students to see the commands most
frequently used in E-STAT. All commands in bold type and selections in bold italic type should be
highlighted.
Resources
Statistics Canada is currently developing an online student activity called Statistics: Power from Data,
which could be an additional resource for the exploration of statistical data. Links for it are Learning
Resources >> List of Learning resources >> Statistics: Power from Data.
Unit 1 – Page 9
 Mathematics of Data Management – University Preparation
Appendix 1.2.1
Sample Responses
Casino Activity (Part A)
From the first graph, it seems that the number of people going to Casinos is on the rise. In 1996, about
1400 people went to a casino; in 1999, about 2750 people went to a casino. The more people casinos
attract, the more money they will make. The bar graph shows the data more clearly. It is easier to
compare the provinces.
The number of Canadians who went to a casino
Statistics Canada – www.statcan.ca or http://estat.statcan.ca
The number of Canadians from Ontario, Quebec, and British Columbia who went to a casino
Statistics Canada – www.statcan.ca or http://estat.statcan.ca
Annual
Canada
Quebec
Ontario
1996
1333
406
454
1997
1786
X
1026
1998
2289
433
1162
1999
2739
487
1439
The X’s indicates that there is data missing which are confidential.
BC
X
X
X
253
Statistics Canada – www.statcan.ca or http://estat.statcan.ca
In Ontario, there is definitely an increase over the years of people attending casinos. This however does
not necessarily mean that casinos are successful in Canada. In Quebec, attendance has not increased. It
looks like in BC casinos are just starting to attract people (this may not be the case since previous data is
confidential).
We need to look at the revenues and costs of casinos to determine if they are successful in Canada. This
data set is not enough to answer the question.
Unit 1 – Page 10
 Mathematics of Data Management – University Preparation
Appendix 1.2.1 (Continued)
Part B, Method 1– Tree Replenishing and Harvesting
Statistics Canada – www.statcan.ca or http://estat.statcan.ca
Highest to Lowest Harvesters: Quebec, Ontario, BC, NB, and NS
Quebec, Ontario, and British Columbia are large provinces that have many forests. The prairie provinces
are large but do not have forests – flat farm land.
Part of the Northwest Territories lies above the tree line – so it is not useful for tree harvesting.
PEI is small and so is the Yukon (which also lies above the tree line).
1997 Data (km2)
Total Area of Trees
Tree Replenishment by Tree Replenishment by
Area
Harvested
Seeding
Planting with Seedlings
Alberta
507
7
453
BC
1758
5
1885
Manitoba
155
62
New Brunswick
1124
200
Newfoundland
200
34
NW Territories
4
0
2
Nova Scotia
695
81
Ontario
1979
219
728
PEI
0
Quebec
3627
17
728
Saskatchewan
175
157
Yukon Territory
0
Statistics Canada – www.statcan.ca or http://estat.statcan.ca
It is clear that the main method of replenishment is by planting seedlings, not seeding.
BC, Saskatchewan, and Alberta roughly replenish the same amount of trees as harvested.
Unit 1 – Page 11
 Mathematics of Data Management – University Preparation
Activity 1.3: Posing Questions, Finding Data, and Critiquing Conclusions
Time: 2 hours
Description
This activity is designed to promote thought and discussion more than simply answering a question.
Students use E-STAT to find data to respond to a statement. Students are assessed on their skills of
navigating through E-STAT to find data, producing graphs, and hypothesizing a possible answer. In small
groups, students present their findings and group members critique the written and oral presentation
using a rubric. The feedback is used to improve the written response. Groups also discuss other factors
that could influence the data or other questions that could be asked.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE5a - works effectively as an interdependent team member;
CGE5b - thinks critically about the meaning and purpose of work;
CGE5e - respects the rights, responsibilities, and contributions of self and others.
Strand(s): Organization of Data for Analysis, Integration of the Techniques of Data Management
Overall Expectations
ODV.01 - organize data to facilitate manipulation and retrieval;
DMV.02 - present a project to an audience and critique the projects of others.
Specific Expectations
OD1.01 - locate data to answer questions of significance or personal interest, by searching wellorganized databases;
OD1.02 - use the Internet effectively as a source for databases;
DM2.01 - create a summary of a project to present within a restricted length of time, using
communications technology effectively;
DM2.02 - answer questions about a project, fully justifying mathematical reasoning;
DM2.03 - critique the mathematical work of others in a constructive fashion.
Prior Knowledge & Skills
 Finding data on E-STAT; creating graphs and tables.
 Analysing and describing trends shown in graphs.
Planning Notes
 Students have seen E-STAT and/or have completed the previous activity.
 Students require individual computer access. If individual access to computers is not possible,
students can work in pairs and the assessment can involve more direct questioning about how to
access the files rather than teacher observation.
 Four questions are provided that have data available on E-STAT; other questions can be developed to
address the particular interests of your class.
 Students become skilled at navigating the Internet to find data; the teacher should observe individual
students’ abilities to follow the instructions from the previous activity.
 Students submit their original and revised presentation and the peer-assessment rubrics.
 In small groups, students have the opportunity to work effectively as a team member and to think
critically about the feedback that the individual provides to the presenter.
 Students write a reflective journal following the presentations to provide constructive criticism of
their work by peers. In their journal, students reflect on their ability to offer constructive criticism to
a peer and on their acceptance of criticism.
Unit 1 – Page 12
 Mathematics of Data Management – University Preparation
Teaching/Learning Strategies
Day 1: Assign four students to a group and provide each group with the four questions to be researched,
providing four different presentations later in groups. Each student works individually at a computer,
using the instructions from the previous activity, to answer their question using data on E-STAT. They
copy their graphs into a word-processing document and respond to the statement. The teacher observes
students’ ability to find the data and provides help as needed. Provide a copy of the rubrics that are used
for observation, peer assessment, and final assessment.
Day 2: In their groups, students take turns presenting their question and their answer. Peer assessment
provides individual students with information on how to improve their presentations. Students generate
other questions that could be asked using the data or other factors that could be investigated that may
have an impact on the data (e.g., population changes, international crises, etc.).
At the end of the group activity, students hand in their original and revised presentations, as well as their
peer assessment rubrics. Presentations could be posted on a bulletin board to help students determine
their ‘significant problem’ for the final assessment.
Student Activity
Part A: Find Data to Respond to the Statement
1. Using E-STAT data and graphs, respond to one of the following statements:
 The ice storm of January 1998 caused extensive damage to the maple trees in Quebec and
Ontario; this damage caused a significant decrease in the production of maple syrup in both
provinces.
 Telephone companies stated that the cost of local service is subsidized by the revenues from
long-distance (toll) calls. With the introduction of long-distance options, prices for local service
increased significantly and now forms a significant proportion of total revenues.
 Employees with a university degree continue to increase their use of the Internet at work and at
home and differ significantly from employees with less education.
 With the change in the Young Offenders Law, there are more kids committing more violent
crimes since they can get off easier.
2. Using available data, select the pertinent series that will help you respond to the statement.
3. Produce line and vertical bar graphs with appropriate titles and copy into a word-processing
document.
4. In your document, respond to the initial statement using the data you found in E-STAT.
Part B: Critiquing Work and Posing New Questions
1. Present your response and graphs to your group.
2. Allow your peers to critique your presentation (Appendix 1.3.2).
3. Discuss the assessment with your peers and record feedback to improve your presentation.
4. As a group, discuss factors that could influence the data (e.g., population, income levels, disease,
etc.). These factors would require more research to be completed before fully responding to the
statement.
5. Generate other questions/statements that could possibly be answered using the same data. Record in
Posing Questions/Problems of your journal.
6. Amend your document to include any improvements and add additional factors that could be
influencing the data.
7. Hand in your initial report and the amended document to your teacher for assessment.
Journal Reflection: In Critiquing, discuss: Did I provide both positive and negative components in my
analysis? Did I accept and use the suggested improvements for my presentation?
Unit 1 – Page 13
 Mathematics of Data Management – University Preparation
Assessment & Evaluation of Student Achievement
This activity is initially assessed by peers, using the Critique Sheet (Appendix 1.3.2). Students make
improvements to their report and submit the original report and the amended report along with questions
or factors that could be investigated. The final report is assessed by the teacher to provide feedback on
use of the Internet, mathematical communication, and critiquing others’ work (Appendix 1.3.1). Initiative
and teamwork can be assessed using a rubric.
Accommodations
 Pair students for appropriate support.
 Provide oral discussion prior to and after the activity to increase the student’s level of
comprehension.
Resources
Statistics Canada, E-STAT – http://estat.statcan.ca
Background information on the Young Offenders act and Violent crimes can be found at:
– http://estat.statcan.ca/content/English/articles/cyb/cyb-just.htm
– http://www.statcan.ca/Daily/English/980127/d980127.htm#ART1
Unit 1 – Page 14
 Mathematics of Data Management – University Preparation
Appendix 1.3.1 – Rubric
Criteria
Level 1
(50-59%)
- demonstrates
limited
understanding
of how to use
the Internet
effectively
Level 2
(60-69%)
- demonstrates
some
understanding
of how to use
the Internet
effectively
Level 3
(70-79%)
- demonstrates
considerable
understanding
of how to use
the Internet
effectively
Level 4
(80-100%)
- demonstrates
thorough
understanding
of how to use
the Internet
effectively
Knowledge
Use the Internet effectively as a
source for databases (OD1.02)
 accesses E-STAT Table of
Contents, selects topic, and
navigates through process
 produces graphs and tables
with relevant titles
- rarely
- sometimes
- justifies
- fully justifies
Problem Solving
Answer questions about a
justifies
justifies
answers with
answers with
project, fully justifying
answers with
answers with
mathematical
complex
mathematical reasoning
mathematical
mathematical reasoning
mathematical
(DM2.02)
reasoning
reasoning
reasoning
- uses
- uses
- uses
- uses
Communication
Create a summary of a project to technology
technology
technology
technology
present within a restricted length with limited
with moderate with
with a high
of time, using communications
effectiveness
effectiveness
considerable
degree of
technology effectively (DM2.01)
effectiveness
effectiveness
Note: A student whose achievement is below Level 1 (50%) has not met the expectations for this
assignment or activity.
Appendix 1.3.2
Critique Sheet
Author of Report:
1. The report is: ( ) excellent ( ) very good ( ) good ( ) fair
2. Information on the visuals (graphs, charts, etc.) is: ( ) thorough ( ) considerable ( ) some ( )
limited
What do you recommend the author do to improve the visuals?
3. Use of analytical methods (consider all the methods studied to date and methods appropriate for this
data) is: ( ) thorough ( ) considerable ( ) some ( ) limited
Give your reasons. (Support for correctly chosen methods and suggestions for omitted methods.)
4. Justification and clarity of conclusions is: ( ) thorough ( ) considerable ( ) some ( ) limited
Justification of reasoning is: ( ) thorough ( ) considerable ( ) some ( ) limited
Give your reasons. (Support what has been done well and provide suggestions for improvement.)
Excerpted from “Problem Posing and Critiquing”, NCTM Journal Mathematics Teaching in the Middle School,
Vol. 4, No. 2, October 1998, pp. 128.
Knowledge
Level 1
Level 2
Level 3
Level 4
Critique the
- demonstrates
- demonstrates
- demonstrates
- demonstrates
mathematical
limited
some
considerable
thorough
work of others in
understanding of
understanding of
understanding of
understanding of
a constructive
critiquing math
critiquing math
critiquing math
critiquing math
fashion (DM2.03) work
work
work
work
Note: A student whose achievement is below Level 1 (50%) has not met the expectations for this
assignment or activity.
Unit 1 – Page 15
 Mathematics of Data Management – University Preparation
Activity 1.4: Using Fathom to Organize Data
Time: 4 hours
Description
Students organize and summarize data from secondary sources by creating a spreadsheet template using
the statistical program, Fathom. This activity can be used as a lead in for several expectations from
Unit 2. In Part A, students learn how to use the analysis features of Fathom, such as least squares
regression line and correlation coefficient. Students review an Internet site addressing cause-and-effect.
The teacher can further develop these Unit 2 expectations using other resources. In Part B, students
merge two different sets of data into one Case Table. In Part C, students make a curve of best fit for
normal data and calculate the mean and median. The teacher can further develop the Unit 2 expectation
of analysing data involving one variable and solving problems involving normal distribution.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE2b - reads, understands, and uses written material effectively;
CGE2c - presents information and ideas clearly and honestly and with sensitivity to others.
Strand(s): Organization of Data for Analysis, Statistics
Overall Expectations
ODV.01 - organize data to facilitate manipulation and retrieval;
STV.01 - demonstrate an understanding of standard techniques for collecting data;
STV.02 - analyse data involving on variable, using a variety of techniques;
STV.03 - solve problems involving the normal distribution;
STV.04 - describe the relationship between two variables by interpreting the correlation coefficient;
DMV.01 - carry out a culminating project on a topic or issue of significance that requires the integration
and application of the expectations of the course.
Specific Expectations
OD1.03 - create database or spreadsheet templates that facilitate the manipulation and retrieval of data
from large bodies of information that have a variety of characteristics;
ST1.04 - organize and summarize data from secondary sources;
ST2.01 - compute, using technology, measures of one-variable statistics, and demonstrate an
understanding of the appropriate use of each measure;
ST2.02 - interpret one-variable statistics to describe characteristics of a data set;
ST3.02 - demonstrate an understanding of the properties of the normal distribution and use these
properties to solve problems;
ST4.01 - define the correlation coefficient as a measure of the fit of a scatter graph to a linear model;
ST4.02 - calculate the correlation coefficient for a set of data, using graphing calculators or statistical
software;
ST4.03 - demonstrate an understanding of the distinction between cause-effect relationships and the
mathematical correlation between variables;
DM1.02 - select an apply the tools of the course to design and carry out a study of the problem.
Prior Knowledge & Skills
 Understanding of mean, median, mode, linear models, and trends in data
 Transformations of functions
 Cause-and-effect when describing relationships
Unit 1 – Page 16
 Mathematics of Data Management – University Preparation
Planning Notes
 Book computers for the entire activity (four hours) and photocopy the activity for students.
 The teacher can teach many expectations from Unit 2.
 In Part A (1 hour), students visit a website dealing with criteria for determining causation
(www.agius.com/hew/resource/assoc.htm). Visit the site ahead of time. Teach cause-and-effect
(Unit 2 expectation) when describing relationships between two variables. Discuss how the
correlation coefficient is not an indicator of cause-and-effect. After Part A is completed, the class
can more thoroughly explore the relationship between two variables (Unit 2 expectation). Part B
(1 hour) reinforces relationships between two variables.
 Briefly introduce normal distribution; students can research normal distribution for homework
from textbook the night before. Part C (2 hours) involves data that can be modelled by a normal
curve using sliders in Fathom. The power of this activity is that students have a visual, created
using technology, of normal distribution. Solving problems involving normal distribution (Unit 2
expectation) can be taught after this activity.
 Use a multimedia projector to introduce Fathom to the class and address common questions (or use
an AVERY key, which is a device that connects a computer to a TV and VCR).
 In their journals students summarize the Data and Story Library (DASL) website, reflect on the moral
and ethical importance of analysing data without bias, and present their ideas clearly and honestly
with sensitivity to others.
 Do this activity beforehand to be able to anticipate students’ questions.
 Impress upon students that this is real data and will not fit the models exactly.
Teaching/Learning Strategies
For the wealth activity, the approximate values for the sliders a, b, k, and d are a = 11, b = - 63.6, k = 0.097, d = 0.93 (these transformation are the same from Grade 11).
Fathom can calculate range, variance, and standard deviation. These Unit 2 expectations could be taught
within this activity. However, student instruction is not given for these measures of one-variable
statistics. It would be helpful to students to do the calculations on the wealth investigation step 4 so that
they have a visual when learning about it in the classroom. To access the tools, right click on graph,
select Plot Value, Functions, Statistical, One Attribute (see Fathom manual).
Journal Reflection: Students briefly describe and reflect on the websites used and pose questions
concerning the data.
Student Activity
Part A: Smoking and Cancer
Finding the Data
1. Use data from Data and Story Library (DASL) http://lib.stat.cmu.edu/DASL/. Follow Links: Data
Subjects >> Health >> Smoking and Cancer (not Smoking and Cancer Datafile). Read what the
file and the variables are about.
2. Using your mouse, start at Number of Cases: 44 and highlight to the end of the explanation of the
fifth variable (Leuk). Copy this by clicking right mouse key or selecting copy from the edit menu.
Do not close the Internet window.
Using Fathom
3. Open Fathom. From the tool bar, select the A tool (text tool). Click on your screen and a text box
appears. Right click and paste text. The box can be made larger by dragging the sides of the box.
Unit 1 – Page 17
 Mathematics of Data Management – University Preparation
4. Return to the Smoking and Cancer site and highlight the data, starting at the beginning of state to the
end of the data table. Right click and copy (Method 1 for retrieving data).
5. Return to Fathom and select the box beside the arrow from the toolbar. Click on your screen and the
box appears. Right click on box and Paste Cases. Objects that look like gold balls should appear in
the box. To name the collection, double click on Collection 1.
6. Double click on the Collection Box. Under comments, record the URL and any links you used to
access the Internet site.
7. Make sure the box is selected (grey outline). From the toolbar, select the table icon (beside box).
Click on screen and the table appears. The data should be in the table of values.
8. Make the table of values box wider so you can see all the headings. Notice that the last two headings
are not properly imported into Fathom. Always verify that Fathom has properly imported your data.
Cleaning Up the Data
9. Double click on KIDLEUK and rename it Kidney. Double click on Attr6 and rename it Leukemia.
10. Verify that the rest of the data is imported correctly. Make both Fathom and the Internet window
smaller so that you can view both at the same time. In Fathom, use the scroll bar to see all the data in
the table of values.
Graphing the Data
11. Maximize space on the Fathom desktop. Make your table of values smaller so that you can only see
the attribute names and the first three cases.
12. Select the graph icon (beside table of values icon) and place on Fathom desktop. Grab the attribute
CIG by holding down on the left mouse button and dragging it over to the graph. Drop it on the xaxis (only the x-axis should have a bolded box around it). Grab the attribute BLAD and drop it onto
the y-axis. The graph should say scatter plot in the top right corner.
13. Select the graph icon again. Place another graph on your Fathom desktop. Compare CIG with LUNG.
Do this for the remaining attributes.
14. Select the A tool and create another textbox. Explain the conclusions you can make from the graphs.
Linear Regressions using Fathom
15. If there is a trend, it is useful to do a linear regression. In Fathom, right click on the graph and select
the Least Squares line. An r^2 value will appear at the bottom of the graph. To determine the
correlation coefficient, take the square root of this value.
Unit 1 – Page 18
 Mathematics of Data Management – University Preparation
Changing Axis Scales
16. The basic way to work with axes in Fathom is to drag on the numbers of the axis. Dragging in the
middle translates the axis, moving the range without changing the scale. Dragging closer to the ends
expands or contracts the range, keeping the opposite end of the axis constant. Think of this action as
zooming in or zooming out (Fathom Reference Manual, p. 20).
17. You can double click on the graph and manually adjust the xmin, xmax, ymin, and ymax. To return
to original scale, select Rescale Graph Axes from the Graph menu.
Tips for Maximizing your Fathom Desktop
 Hide items you are not using (e.g., the box with the gold balls). Select the item and from Display
select Hide Collection. If you need to see it again, select from Display Show Hidden Objects (this
will show all hidden objects).
 When finished with a table of values and only the graphs are important, delete the table of values and
the graphs remain. To retrieve the table of values, select the collection box (with gold balls), select
the table of values from the toolbar, and bring it onto the Fathom desktop.
In Your Notebooks: Go to www.agius.com/hew/resource/assoc.htm. List the 11 criteria for determining
causation. Discuss each criterion in reference to the smoking and cancer data from DASL. Discuss the
research that is needed to be sure that there is a cause-and-effect relationship between smoking and
cancer. Return to the website where the data is from to get additional information. Also discuss the role
of the correlation coefficient and cause-and-effect relationships. Write your report within the Fathom
document.
Journal Reflection: In Ethical Implications of Data, reflect on the moral and ethical implications of
smoking and how they relates to this data. In Posing Questions/Problems, create a question related to
cancer and/or smoking that could be investigated by analysing data.
Part B: Sport Injuries in Football
1. Access www.unc.edu/depts/nccsi/ >> Data Tables >> Annual Survey of Football Injury
Research.
2. Copy the table Fatalities Directly Due to Football.
3. Open Fathom. Paste cases into a Collection Box (Method 1).
4. Create a Case Table and observe the data. We need to do a little clean up. Create a scatter plot.
5. The first two cases cause problems when graphing. Holding shift, select Case 1 and Case 2; right
click and select Delete Cases.
6. Delete the last three attributes since there is no data in the columns. Holding shift, select the last
three columns; right click and select Delete Attributes.
7. Relabel the attributes to the appropriate headings. Minimize the Fathom screen so that you can see
both Fathom and the Internet page.
8. Relabel the collection Football Fatalities by double clicking on the name Collection 1.
9. Double click on the Collection Box and record the URL in the comments section and the name of the
site. Do this for all data that you use.
10. Create a scatter plot for Year and High School Fatalities.
11. Notice it does not produce a scatter plot because the last case has the word TOTAL. Delete the last
case and create a new scatter plot. Create the least squares line.
Unit 1 – Page 19
 Mathematics of Data Management – University Preparation
In Your Notebook: Discuss the possible reasons for this trend and discuss the correlation coefficient.
Importing a Second Table of Data from a Different URL
12. Return to the website and press Back. Select Annual Survey of Catastrophic Injuries.
13. Copy the table Cervical Cord Injuries.
14. On the same desktop as Football Fatalities, select a new Collection Box and paste cases.
15. Create a Case Table and clean the table up. Relabel this collection Cervical Cord Injuries in Football.
Create a scatter graph of Year and High School.
In Your Notebook: Discuss the possible reasons for this trend and compare it to the football fatalities.
Notice the scale of each graph.
Merging Case Tables
16. Create a new Case Table. Create the attribute labelled x.
17.
18.
19.
20.
Notice a new Collection Box is automatically created.
Compare football fatalities with cervical cord injuries for each year.
Relabel the Collection Football Fatalities compared to Cervical Cord Injuries.
Click on Year from the Football Fatalities Case Table. The column should be highlighted. From the
Edit Menu, select Copy Attribute. Click on x. From the Edit Menu, select Paste Attribute. Copy the
high school Attribute and paste it in y. Relabel this attribute Secondary School Fatalities (use an
underscore for a space).
21. Since Cervical Cord Injuries only begin in 1977, delete all cases prior to 1977 in the Football
Fatalities compared to Cervical Cord Injuries Case Table.
22. Copy and paste high school attribute from Cervical Cord Injuries Case Table. Relabel it.
23. Create a scatter plot comparing Football Fatalities and Football Cervical Cord Injuries.
In Your Notebook: Discuss the criteria for determining causation with reference to this data for sports
injuries. What additional information is needed for determining if this is a cause-and-effect relationship?
What does the correlation coefficient tell us about this data?
Journal Reflection: What are the moral and ethical implications of this relationship? In Posing
Questions/Problems, create a question related to Sport Injuries that could be investigated by analysing
data.
Part C: Investigating Wealth
1. Use data from DASL: Data Subjects >> Economics >> Billionaires 92 Datafile.
2. Import Data into Fathom (Method 2). Right click on URL and copy. In Fathom, from File Menu,
select Import from URL. Right click on address box and select paste. Select OK. A case box should
appear with Gold Balls in it. Select table of values and check to see if data was imported. If not, use
Method 1 from the cancer activity.
3. Create a table of values on the desktop. Graph age versus wealth. Notice that the graph says Dot Plot
at the top right corner and does not have a scale for the x-axis. To make a scatter plot, there must only
be numbers in the table of values. If you scroll down to case 105, there is an asterisk in the cell. This
is another type of data clean up that you will have to check for. Delete all asterisks and delete the
graph.
Analysing One-Variable Data
4. Select graph and place on desktop. Drag age and put it on the x-axis. Notice the shape of the dot plot.
This looks to be a normal distribution. To verify, calculate the mean and median of the data. If they
are equal, then it is a normal distribution.
5. Right mouse click on graph. Select Plot Value. Type on the screen mean(age). Press OK. Repeat this
process to calculate the median. Notice they are approximately the same and they are in the middle of
the dot plot.
Unit 1 – Page 20
 Mathematics of Data Management – University Preparation
6. Change this graph to a histogram by clicking on Dot Plot and selecting Histogram. It is easy to see
the mode if you double click on the x-scale and change bin width to 1. The mode is 68; this data is
not exactly normal. To return to the original graph, select Rescale Graph from Graph Menu.
Using Sliders to Analyse Data
7. Create another graph on the desktop and put age on the x-axis and wealth on the y-axis.
8. Bring down 4 sliders from the toolbar (the icon beside the A). Label a, b, k, and d.
9. Right click on the graph. Select Plot Function. Click on + sign beside Function. Click on + sign
beside distribution. Click on + sign beside Normal. Double click on normalDensity (the description
of this tool is at the bottom of the Expression for function screen).
10. Type in the letters a, k, b, and d. Select OK.
11. Using the sliders, try to fit a curve of best. Drag the slider on the scale to change the value of a, b, k,
and d.
Changing a Slider’s Scale: Use the same rules as Changing Axis Scale
Journal Reflection: In Ethical Implications of Data, reflect on how this data could be useful: What is this
data telling us? In Data Sites, write a summary concerning the DASL website. In Posing
Questions/Problems, create a question related to wealth that could be investigated by analysing data.
Assessment & Evaluation of Student Achievement
 Focus on formative assessment (i.e., giving informal feedback to students). Activity 1.5 is the
assessment activity that complements this activity.
 Use peer assessment to provide feedback for cause-and-effect entry in notebook.
 Collect journals to provide feedback on the questions posed. Are they too narrow or too open?
 Learning skills could be assessed in this activity.
Resources
Fathom (information and teacher resources) – www.lat-olm.com.au/Fathom.htm.
Exploring Data (introductory statistics) – http://exploringdata.cqu.edu.au/ .
Information Literacy Skills – http://score.rims.k12.ca.us/infolit.html .
Activity 1.5: Finding Data to Answer Questions Using Fathom
Time: 4 hours
Description
This activity is an assessment activity of the skills and techniques learned in Activity 1.4. Students are
given a question or statement and the URL for the data site. Students extract information from the sites
and import the data into Fathom; manipulation of data may be required. Students analyse the data and
produce a report responding to the initial question and suggesting other factors that could influence the
observed data.
Unit 1 – Page 21
 Mathematics of Data Management – University Preparation
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE2b - reads, understands, and uses written materials effectively;
CGE2c - present information and ideas clearly and honestly and with sensitivity to others;
CGE3d - makes decisions in light of gospel values with an informed moral conscience;
CGE4e - applies effective communication, decision-making, problem-solving, time, and resource
management skills.
Strand(s): Organization of Data for Analysis, Statistics, Integration of the Techniques
of Data Management
Overall Expectations
ODV.01 - organize data to facilitate manipulation and retrieval;
STV.01 - demonstrate an understanding of standard techniques for collecting data;
DMV.01 - carry out a culminating project on a topic or issue of significance that requires the integration
and application of the expectations of the course;
DMV.02 - present a project to an audience and critique the projects of others.
Specific Expectations
OD1.03 - create databases or spreadsheet templates that facilitate the manipulation and retrieval of data
from large bodies of information that have a variety of characteristics;
ST1.04 - organize and summarize data from secondary sources using technology;
DM1.02 - select and apply the tools of the course to design and carry out a study of the problem;
DM2.03 - critique the mathematical work of others in a constructive fashion.
Prior Knowledge & Skills
 Finding data on the Internet; creating graphs and tables using Fathom
Planning Notes
Data sites have been provided to allow quick access to the data; students need time to import, clean up
and merge data and then produce their report. This activity is an assessment of the student’s ability to
complete these actions. Students are not expected to thoroughly investigate the question. Students may
discover that they have more questions than answers; students become more aware of relevant factors
that could affect the data or of alternate ways in which they must analyse the data.
Suggestions
Question 1: Only three time periods are included; students should realize that it is a weak data site. Be
sure that students look at daily smokers.
Question 2: For Air – choose PASSENGER TRAFFIC, by Sector; Rail – choose PASSENGER
SERVICE, Total Passengers; Bus – choose INTERNATIONAL COMPARISONS, Intercity
Passengers; Cars – choose VEHICLES, Registrations by Region. To make comparisons, all data
should be expressed as 000’s. In Fathom, create a <new> attribute and insert formulas that
multiply/divide the required attribute by the appropriate value. Delete extra attributes and cases (19881999 are common to all tables). Students should address the validity of a linear relation between air
travel and car registrations, which requires the merging of four tables from the one site.
Question 3: Requires the merging of two tables that are produced from different sites; students need to
recognize the inverse nature of income level of Canadians vs. year and poverty percentage versus year;
other graphs are possible.
Question 4: Looks at a simple normal distribution.
Unit 1 – Page 22
 Mathematics of Data Management – University Preparation
Teaching/Learning Strategies
Students complete individual work on the computers to produce a written assignment. If individual
access is possible, students should complete their assignment on the computers. If time is restricted,
students can complete the data retrieval and analyses on the computer but complete the assignment by
hand. If individual access to computers is not possible, students can work in pairs to retrieve data and
complete analyses and then finish the assignment by hand, individually.
The teacher observes students’ ability to retrieve data from the Internet and to use Fathom software for
analysing data. If students worked in pairs, they should exchange their papers with a different student for
the peer critique. Students critique the work of a classmate using the rubric in Activity 1.3. Students then
submit their work for a summative assessment and the critique of a student’s work for formative
assessment. In their critique, students present their ideas clearly and honestly and with sensitivity to
others. In their journals, students reflect on their reaction to the quality of the other student’s work and on
their ability to offer constructive criticism to a peer.
Student Activity
Part 1: Start with a Question and Search for the Data
There are four questions to answer. Find each of the data sites through the identified link. Choose the
appropriate descriptors to extract the required data and import the data into Fathom. Create case tables,
merge attributes from two tables into one new table, and add formulas to allow for comparisons of data
(e.g., 1’s to 000’s). Create one or more graphs that support or refute the question. Prepare a report that
includes the data, the graph(s), and a short summary that outlines the techniques used to prepare the
report. Analyse the data using skills learned in previous lessons and write a conclusion. Discuss other
factors that could have an influence on the data and explain your reasons.
 Have the education and advertising campaigns addressing the harmful effects of smoking caused a
decrease in the total number of adolescent daily smokers across Canada? Has the change been more
significant in males or females? E-STAT >>Search CANSIM for Table 104-0027.
 How do Canadians travel within their own country? As more people buy more cars, has the rail
system (passengers) and bus companies (intercity travel) noticed a decrease in the number of
travellers? Can the number of car registrations (highways) be used to predict the air passengers
(http://www.tc.gc.ca/en/menu.htm – passenger traffic by sector, Transport Canada). Using the menu
across the top of the page, drop down menu on Road and select More, then look on the left and select
“Statistics and forecasts,” scroll down to Statistical Data. Use the terminology indicated to find the
relevant data.
 Canada is a wealthy nation but many children continue to live in poverty. Is the income level of
Canadians a factor that affects the percentage of child poverty in Canada? (Search, data child poverty
in Canada) www.ccsd.ca/cpovhist.htm (Child Poverty Rates, Canada) and http://estat.statcan.ca/,
Personal finance and household data, Table for income based on selected family type in Canada
 Flies are all the same, aren’t they? Is there any variation in the lengths of housefly wings?
www.seattlecentral.org/qelp/index.html
Part 2: Peer Critiques
Exchange your assignment with another student. Use the Critique Sheet to critique your classmate’s
work.
Hand in the critiqued assignment, along with the critique, to the teacher for separate assessments.
Journal Entry: In Ethical Implications of Data, discuss your reaction to the data and articles on child
poverty. How can we change this situation in Canada? In Critiquing, reflect on your reaction to the
quality of work that you saw and on your ability to offer constructive criticism to a peer.
Unit 1 – Page 23
 Mathematics of Data Management – University Preparation
Assessment & Evaluation of Student Achievement
 Summative assessment of written report using rubric
 Formative assessment of the critique
Resources
Statistics Canada – http://estat.statcan.ca
Transport Canada – www.tc.gc.ca/en/menu.htm
Child Poverty – www.ccsd.ca/cpovhist.htm
Data and Story Library – http://lib.stat.cmu.edu/DASL/
Child Poverty 2000 – www.campaign2000.ca/NATrc00.pdf
Quantitative Environment Learning Project – www.seattlecentral.org/qelp/index.html
Unit 1 – Page 24
 Mathematics of Data Management – University Preparation
Appendix 1.5.1
Rubric
Knowledge
Use the Internet
effectively as a
source for
databases
(OD1.02)
Organize and
summarize data
from secondary
sources using
technology
(ST1.04)
Level 1
(50-59%)
- demonstrates
limited
understanding of
how to use the
Internet
effectively
- demonstrates
limited
understanding of
organizing and
summarizing data
Level 2
(60-69%)
- demonstrates
some
understanding of
how to use the
Internet
effectively
- demonstrates
some
understanding of
organizing and
summarizing data
Level 3
(70-79%)
- demonstrates
considerable
understanding of
how to use the
Internet
effectively
- demonstrates
considerable
understanding or
organizing and
summarizing data
Level 4
(80-100%)
- demonstrates
thorough
understanding of
how to use the
Internet
effectively
- demonstrates
thorough
understanding of
organizing and
summarizing data
Activity 1.6: Looking Critically at Websites
Time: 2 hours
Description
Students evaluate the credibility of websites. In Activity 1.1, students were given the assignment How to
Determine if a Website is Credible; students designed a checklist to identify the strengths, shortcomings,
biases, validity, etc. of different websites. Students use their checklists to examine different websites that
may be useful as data sources for their final project.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE1d - develops attitudes and values founded on Catholic social teaching and acts to promote social
responsibility, human solidarity, and the common good;
CGE3e - adopts a holistic approach to life by integrating learning from various subject areas and
experience;
CGE5a - works effectively as an interdependent team member;
CGE5e - respects the rights, responsibilities, and contributions of self and others.
Strand(s): Organization of Data for Analysis; Statistics
Overall Expectations
ODV.01 - organize data to facilitate manipulation and retrieval;
STV.01 - demonstrate an understanding of standard techniques for collecting data.
Specific Expectations
OD1.02 - use the Internet effectively as a source for databases;
ST1.04 - organize and summarize data from secondary sources using technology.
Prior Knowledge & Skills
 Knowledge of the dynamic statistics software Fathom.
Unit 1 – Page 25
 Mathematics of Data Management – University Preparation
Planning Notes
 Book computers with Internet access for the activity.
 Reserve multimedia projector for display of checklists.
 Read and do the assignment on Internet Site Credibility.
 Require that students have their checklists on a disk for the discussion.
 Students use the attitudes and values developed by Catholic social teaching when discussing the
motivation of Internet websites.
Teaching/Learning Strategies
Students display their checklists from the completed assignment. The teacher leads a discussion of the
completed assignment and the checklists during which the class develops the final checklist for this
activity. Students record the checklist in their journals.
This activity follows a jigsaw strategy. The teacher organizes students into Home groups of five students.
In the Home groups, students select an Internet site from the list. Students form their Expert groups
according to the site they have chosen to review. Students return to their Home groups and share
interesting observations and information the Expert group found important.
Assignment: How to Determine if a Website is Credible
Use a tutorial created by the University of Winnipeg to create a checklist for critiquing the credibility of
Internet sites. Follow the links to access the tutorial: http://cybrary.uwinnipeg.ca/ >> EMANUEL or
learning about information >> 8: Evaluation the Weakest Link >> Evaluation Criteria.
Answer the following questions.
1. Why is print material considered more credible than Internet material? (Robert Harris “Evaluating
Internet Research Sources”).
2. According to Robert Harris what kind of Information Exists on the Internet?
3. What tip does Robert Harris offer to determine if a source is reliable/credible?
4. What test is a single perfect indicator of reliability, truthfulness, or value? For the Credibility,
Accuracy, Reasonableness, Support (CARS) checklist?
5. Summarize the CARS checklist. Include important questions you must ask yourself and indicators of
poor information when evaluating an Internet site for each of the topics.
6. How can you tell the motivation and source of a document from the Internet address? Is this a strong
indicator of motivation?
7. Using Robert Harris’s article and The University of Winnipeg’s Evaluation – The Weakest Link
Page, create a checklist for evaluating an Internet site.
Student Activity: Evaluating Websites
1. The teacher assigns you to a Home group. In your Home group, each person chooses an Internet site
to review using the checklist developed in class as a guide.
2. Evaluate your chosen site using your checklist, recording any interesting observations and sharing
this information with the other students in your Expert group.
3. Return to your Home Group; report on your reviewed site and listen to the reports of the other sites.
4. Following this sharing of information, search for an Internet data site and complete a review.
5. Return to your Home group and report on the site and listen to the reports of other sites.
6. Choose one site to share with the class.
7. Hand in your review to be posted for class use.
8. Complete the journal entry.
Journal Reflection: In Ethical Implications of Data, write a brief description and address the credibility
of each of the websites presented. List the strengths and weaknesses of each site. Use the websites as
examples to reflect on the moral and ethical importance of analysing data without bias. In Final Project
Brainstorm, record any data sites that you may want to use for your project.
Unit 1 – Page 26
 Mathematics of Data Management – University Preparation
Sites to Review in Expert Groups
 Environment Canada – www.ec.gc.ca/Ind/ >> English.
 Transport Canada – www.tc.gc.ca/ English >> place your arrow on Rail and wait for drop down
menu, select [More …] >> Statistic’s and forecasts >> Scroll Down to Statistical Data.
 Economagic.com: Economic Time Series Page – www.economagic.com. Data cannot be copied and
pasted directly. Select Display series in COPY/PASTE format. Not all data is shown. You must be a
subscriber to access the full dataset. Fathom Tip: Data is shown by years and then by months and
cannot produce a graph in Fathom (try to!!). To graph this over the years, create a new attribute and
label it Time. Right click on Time and select Edit Formula. Enter the formula Attr1 + (Attr2 – 1)
*1/12. This combines months and year.
 US Naval Observatory – http://aa.usno.navy.mil/data/
 The World Bank – www.worldbank.org/data/
To find a data website to review on your own:
1. Search for Data on a search engine.
2. Try the Fathom site: http://www.keypress.com/Fathom >> Links to Data >>.
Assessment & Evaluation of Student Achievement
 Collect student website critiques journals and provide formative assessment feedback.
 Learning strategies could be assessed in this activity.
Accommodations
This activity has a moderate density of words and instructions. To ensure all students understand the
instructions, the teacher should read them to the class as a whole.
Resources
Explanations of jigsaw teaching strategy can be found at the following sites:
– www.discover.tased.edu.au/english/strategy.htm
– www.broward.k12.fl.us/ci/whatsnew/strategies_and_such/strategies/jigsaw.html
Activity 1.7: Culminating Activity for Unit 1
Time: 5 hours
Description
This culminating activity is a scaled-down version of the culminating project. Students select a topic;
find secondary data and supporting data to investigate their topic; evaluate the sites; import the data into
Fathom; do analysis; and form conclusions. They use a poster to display their analysis and conclusions.
Students then critique the posters of their peers, applying effective communication and responsible
decision-making.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE3b - creates, adapts, and evaluates new ideas in light of the common good;
CGE3e - adopts a holistic approach to life by integrating learning from various subject areas and
experience;
CGE4e - sets appropriate goals and priorities in school, work, and personal life;
CGE4f - applies affective communication, decision-making, problem-solving, time, and resource
management skills.
Unit 1 – Page 27
 Mathematics of Data Management – University Preparation
Strand(s): Organization of Data for Analysis, Integration of the Techniques of Data Management
Overall Expectations
ODV.01 - organize data to facilitate manipulation and retrieval;
DMV.01 - carry out a culminating project on a topic or issue of significance that requires the integration
and application of the expectations of the course;
DMV.02 - present a project to an audience and critique the projects of others.
Specific Expectations
OD1.01 - locate data to answer questions of significance or personal interest, by searching wellorganized databases;
OD1.02 - use the Internet effectively as a source for databases;
OD1.03 - create database or spreadsheet templates that facilitate the manipulation and retrieval of data
from large bodies of information that have a variety of characteristics;
DM1.01 - pose a significant problem whose solution would require the organization and analysis of a
large amount of data;
DM1.02 - select and apply the tools of the course to design and carry out a study of the problem;
DM1.03 - compile a clear, well-organized, and fully justified report of the investigation and its findings;
DM2.03 - critique the mathematical work of others in a constructive fashion.
Prior Knowledge & Skills
 Knowledge of Fathom or statistical software.
Planning Notes
 Book computers with Internet access for the entire activity.
Teaching/Learning Strategies
Student Activity
Investigate a topic of your choice and produce a poster displaying your report.
 Pose a Significant Problem whose solution would require the organization and analysis of a large
amount of data. Your problem should be broad in scope and consider a variety of factors that might
influence the outcome of your conclusions. You may use data found in a previous activity.
 Use the Internet to find a large amount of data to analyse your problem. You must evaluate the
credibility of your site and include this evaluation in the bibliography.
 Organize the Data by importing into Fathom. Look at the trends in the data and state conclusions
concerning the data.
 Find Supporting or Influencing Data or Information concerning your problem (you could import data
into Fathom, use E-STAT graphs, include an article, etc.).
 Compile a clear, well-organized, and fully justified report of the investigation and your findings to be
summarized and displayed on a poster. Include a complete bibliography and a reflection piece on the
moral and ethical implications of the data and your conclusions.
 Critique the mathematical reports of your peers in a constructive fashion.
Assessment & Evaluation of Student Achievement
Provide formative feedback for the course project. The assessment rubric consists of both formative and
summative assessment. Feedback should be provided so that students understand what type of question is
significant for their Culminating Project (Unit 5). The other part of the rubric is summative since students
have already been assessed for using the tools of the course, their reasoning, and their critiquing skills.
Accommodations
Students could use alternative strategies to present their projects (e.g., oral, PowerPoint).
Unit 1 – Page 28
 Mathematics of Data Management – University Preparation
Appendix 1.7.1
Formative Assessment
Level 1
(50-59%)
Criteria
Thinking/
Inquiry and
Problem
Solving
DM1.01
- poses a problem
of limited scope
- allows for limited
connections to
other influencing
factors
Level 2
(60-69%)
Level 3
(70-79%)
- poses a problem
of some scope
- allows for few
connections to
other influencing
factors
- poses a problem
of considerable
scope
- allows for some
connections to
other influencing
factors
Level 4
(80-100%)
- poses a problem
of broad scope
- allows for many
connections to
other influencing
factors
Summative Assessment
Expectation
Application –
selection and
sequencing of
tools
DM1.02
ODV.01
Level 1
(50-59%)
Level 2
(60-69%)
Level 3
(70-79%)
Level 4
(80-100%)
- rarely selects and
sequences
appropriate tools
(Internet site
critique, data
organization tool,
data finding
technique)
- sometimes
selects and
sequences
appropriate tools
(Internet site
critique, data
organization tool,
data finding
technique)
- communication
is generally
understandable
and complete
- some correct use
of mathematical
forms
- most often
selects and
sequences
appropriate tools
(Internet site
critique, data
organization tool,
data finding
technique)
- communication
is generally clear
and complete
- always or almost
always selects and
sequences
appropriate tools
(Internet site
critique, data
organization tool,
data finding
technique)
- communication
is consistently
clear and complete
- communication
is limited and
rarely clear and
complete
- limited use of
- generally correct - consistent
correct
use of
correct use of
mathematical
mathematical
mathematical
forms
forms
forms
- judges the
- judges the
- judges the
- judges the
Thinking/
validity of
validity of
validity of
validity of
Inquiry and
conclusions
with
conclusions
with
conclusions
with
conclusions with a
Problem Solving
DM2.03
limited
moderate
considerable
high degree of
effectiveness
effectiveness
effectiveness
effectiveness
Note: A student whose achievement is below Level 1 (50%) has not met the expectations for this
assignment or activity.
Communication
DM1.03
Unit 1 – Page 29
 Mathematics of Data Management – University Preparation
Unit 5: Managing the Culminating Project
Unit Description
Students prepare to successfully complete the culminating project outlined in the Integration of the
Techniques of Data Management strand. Students engage in several activities in which they apply several
of the techniques/tools of the course to answer significant questions. Each activity could be viewed as a
mini-project, providing the teacher with a vehicle for giving each student an opportunity to prepare a
written report, make a presentation to the class, and have it critiqued by other students. The student gains
valuable experience with these three expectations that form part of their culminating project.
Activity 5.1: Stages of the Culminating Project
Time: 8 hours [in addition to the hours allocated within Activities 5.2 to 5.6]
Description
This activity is a series of small activities, mini presentations, checklists, and timing supports designed to
guide students through a process to complete the culminating project. The time allocated is spread
throughout the course. There are several opportunities for students to make presentations and receive
feedback. However it is expected that different students have opportunities in each case. Students benefit
from discussion and feedback after the presentations as they apply the ideas to their own project
development. The planning and implementation process has been broken down into five stages that align
with the units of the course.
Stage 1: (1 hour) During Unit 1, students select a topic and establish a list of significant questions to
investigate for the culminating project. At the end of the unit they should submit their proposals.
Stage 2: (3 hours) During Unit 2, students collect more data and begin analysis using statistical tools.
Students may work through guided Activities 5.2 and 5.3. Presentations and reports may be prepared by
individuals or groups; discussion and feedback provide guidance.
Stage 3: (2 hours) Time allocated to apply the learning from Unit 3 to work with culminating projects.
Students work through guided Activity 5.3 and write reports; selected students present their work.
Feedback and discussion assist students in making progress with their culminating projects.
Stage 4: (1 hour) During Unit 4, students work with additional tools if needed. Students should be
reaching the final stages of their projects.
Stage 5: (1 hour) Time is allocated for students to finish their reports.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE1i - integrates faith with life.
Overall Expectations
DMV.01 - carry out a culminating project on a topic or issue of significance that requires the integration
and application of the expectations of the course.
Specific Expectations
DM1.01 - pose a significant problem whose solution would require the organization and analysis of a
large amount of data;
DM1.02 - select and apply the tools of the course to design and carry out a study of the problem;
DM 1.03 - compile a clear, well-organized, and fully justified report of the investigation and its findings.
Prior Knowledge & Skills
 Stages of the planning process should be examined at the completion of each unit of the course.
Unit 5 – Page 1
 Mathematics of Data Management – University Preparation
Planning Notes
 When planning timelines for the course, it is important to build in the time allotted for these
management activities. Some of these activities are done with the class; individual or group
conferencing may be necessary to keep students on track. Teachers may conference with each student
on a regular basis, so that the process remains ongoing. To allow time to for conferences, set aside
class time for students to work on their projects at midpoint and end of each unit. During those
classes, schedule short conferences with students to review the status of their projects. A tracking
sheet for students and teachers (see Appendix 5.1.1) might serve as the initial page in a portfolio.
 Depending on the size of the class, individual projects and presentations may not be feasible. As
early in the course as possible, teachers need to decide if students may work on a culminating project
individually, in pairs, or in small groups. Some topics may be larger issues than anticipated and it
may be appropriate for two or more students to examine several questions arising from the same
topic. If students are working on a culminating project together, it is important to establish clear
expectations. Each student involved is expected to apply the skills and tools of the course to his/her
part of the project, prepare a report of the part, and present the part to the class. Students may work
on different aspects of the same issue, using the same or different sets of data, but doing their
individual analyses.
 A student portfolio would facilitate student/teacher conferences at the various stages. Students
include data and the sources of the data in their portfolios, helping teachers to assess whether the
proposal is feasible and helping students focus or redefine their questions where necessary.
Teaching/Learning Strategies
Stage 1: Posing a significant problem that is the basis for the culminating project
Students choose an area of interest. Brainstorming project ideas in class may help students make a
choice. Teachers stress that students need to choose a topic of personal interest. Ideally, they will be
investing a lot of time and effort into this culminating project and it is important that they “own” the idea.
Students do a preliminary search for data before finalizing their choice of topic and posing the questions
that they intend to answer. Teachers should consult the teacher-librarian about helping students perform a
proper web search. Students may have trouble locating suitable data; they may need to redefine their
questions or choose a different area of study. In other cases, new questions or areas of interest may
surface from the preliminary search for data. A data search may reveal that the question posed is too big
an issue or there are too many factors involved. It may be necessary to narrow or refine the question at
various stages in the course.
The teacher plays the role of facilitator throughout this process and provides feedback. Other teachers
and guest speakers could be used to open students’ eyes to how research affects our lives on and off the
job.
Once students are satisfied that there is sufficient data, they submit a proposal for the project in writing.
Teachers provide a deadline for proposals. The proposal should include a hypothesis based on the
student’s data search. Some class time should be spent developing hypothesis statements. A sample
proposal sheet is included as (see Appendix 5.1.1). A rubric could be used to assess the proposal.
(See Appendix A.)
Stage 2: Applying data analysis in the culminating project
After the completion of Unit 2: Data Analysis, students revisit their culminating projects and apply the
acquired skills and concepts where appropriate. Since the culminating project must involve the
organization and analysis of a large amount of data, the skills of this unit must be part of all culminating
projects.
Unit 5 – Page 2
 Mathematics of Data Management – University Preparation
A checklist of questions may help students with this process:
 Is the data you have collected pertinent to your project?
 How valid is the data?
 What sampling techniques were used to collect the data?
 Is there possible sampling bias and/or variability?
 Have you organized the data in a way that facilitates its manipulation and retrieval?
 Have you computed the measures of one-variable statistics (mean, median, mode, range, interquartile
range, variance, standard deviation) where appropriate?
 Have you included z-scores and percentiles where appropriate?
 Have you chosen a regression that models the relation between two variables?
 Have you described the relation between two variables by interpreting the correlation coefficient?
 Can the normal distribution be applied with the data in your project?
After the study of one-variable statistics, it may be appropriate to introduce Activity 5.2: Income in
Canadian Families to give students an opportunity to work through a guided example in how one-variable
statistics might be used. It would also be an opportunity for some students to practise writing a report and
making a presentation. At the end of Unit 2, students could work on Activity 5.3: AIDS in Canada as an
example of two-variable statistics and an introduction to the concept of a simulation. Select group of
students could be asked to make a presentation.
Stage 3: Applying counting and probability in the culminating project
Probability concepts and simulating and predicting will not necessarily apply to all culminating projects.
After the completion of Unit 3: Counting and Probability, students should revisit their culminating
projects and apply the acquired skills and concepts. A checklist of questions may help students with this
process:
 Can permutations and combinations be applied in your project?
 Is it possible to consider probability problems associated with the data in your project?
 Can empirical probabilities be calculated and would this be appropriate in the context of your
project?
 Is it possible and appropriate to determine expected values in the context of your project?
 Is it appropriate to construct and use a probability distribution with your data?
 Is it possible to design a simulation as part of your culminating project?
 If a simulation is possible, have you assessed the validity of the simulation results?
At the end of Unit 3 students could work on Activity 5.4: Dice Games. (Parts of Activity 5.4 could also
be used as an introduction to probability concepts, wrapping the activity up at the end of the unit.) A
selected group of students could make presentations.
Stage 4: Applying additional tools for data management in the culminating project
After the completion of Unit 4: Additional Tools for Data Management, students should revisit their
culminating projects:
 Have you included diagrams where appropriate?
 Does graph theory apply to your culminating project?
 Can matrix tools be applied to your culminating project?
Students should now be working towards finishing their final project.
Unit 5 – Page 3
 Mathematics of Data Management – University Preparation
Stage 5: Preparing the report
Students should be aware of the expected components of their report. A possible list might be:
 Cover page including a title that makes the purpose of their project apparent;
 A clear statement of the question to be considered;
 Description of procedure;
 Presentation of data using tables, charts, graphs;
 Summary statistics;
 Evidence of the use of technology;
 Analysis of data including calculations;
 Conclusions;
 Evaluation of your techniques;
 Bibliography.
Students should be aware that their report is to be assessed for its mathematical validity. The
mathematical content of their report should be substantial. This project is their opportunity to
demonstrate their understanding of the skills and concepts of this course in an integrated approach.
Develop a rubric with students (see Appendix B) for assessing the mathematical content of their reports.
Assessment & Evaluation of Student Achievement
It is important that assessment strategies address the process as well as the final product for the
culminating project. Conferencing with students and assessing the various stages in the planning process
are useful for formative assessment. Assessment tools, such as checklists, portfolios, and rubrics, are
useful.
Accommodations
 Students with weak time-management skills may need closer monitoring. A calendar with clearly
indicated deadlines or meeting dates or contracts may be useful for setting due dates for the various
stages of the culminating project process. Students should be encouraged to determine their own
window for due dates.
 Teachers should refer to Individual Education Plans (IEPs) in place for their students in need of
accommodations (e.g., some students may need increased time to complete tasks, while others may
need more frequent conferencing to help them through the process).
Unit 5 – Page 4
 Mathematics of Data Management – University Preparation
Appendix 5.1.1
Student Worksheets
Worksheet 1: Keeping on Track
Student:
Presentation Date:
Stage
Proposed Date
of Completion
Completion
Conference Notes
1. Proposal
2. Data Collected
3. Data Analysis
One-Variable
Two-Variable
4. Other Tools
5. Conclusions
6. Written Report
Planned summary
7. Presentation
Outline
Technology
Timing Checked
Presentation Date:
Worksheet 2: Student Proposal
Due Date:
Name:
(If you are working in a group, include the names of other members of your group who are addressing the
same topic.)
The area of study I intend to investigate is:
Why did you choose this particular topic?
Do you have an expectation about the results you will find?
I have done a preliminary search for data and feel that the data is appropriate and sufficient for the
analysis that will be necessary. Based on this preliminary data, the question I will consider is:
Unit 5 – Page 5
 Mathematics of Data Management – University Preparation
Activity 5.2: Income in Canadian Families – A One-Variable Data Activity
Time: 4 hours [use suggested at or near the end of Unit 2]
Description
In this open-ended activity, with opportunity for further exploration, students analyse data about family
income in Canada over a 20-year period; they then use their analyses to pose and answer questions.
Students organize data from the Internet, creating suitable intervals and a frequency table over several
years, graph the frequency distributions, and discuss changes and trends. Comparisons of the mean,
median, standard deviation, and quartiles are used to describe trends in the data [before and after
adjusting the income figures using the Consumer Price Index (CPI) which they download from the
Internet]. Z-scores and percentiles are used to describe individual pieces of data.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE3c - a reflective and creative thinker who thinks reflectively and creatively to evaluate situations and
solve problems.
Overall Expectations
ODV.01 - organize data to facilitate manipulation and retrieval;
STV.01 - demonstrate an understanding of standard techniques for collecting data;
STV.02 - analyse data involving one variable, using a variety of techniques;
STV.05 - evaluate the validity of statistics drawn from a variety of sources.
Specific Expectations
OD1.01 - locate data to answer questions of significance or personal interest, by searching wellorganized databases;
OD1.03 - create database or spreadsheet templates that facilitate the manipulation and retrieval of data
from large bodies of information that have a variety of characteristics;
ST1.04 - organize and summarize data from secondary sources using technology;
ST2.01 - compute, using technology, measures of one-variable statistics and demonstrate an
understanding of the appropriate use of each measure;
ST2.02 - interpret one-variable statistics to describe characteristics of a data set;
ST2.03 - describe the position of individual observations within a data set, using z-scores and percentiles;
ST5.03 - explain the meaning and the use in the media of indices based on surveys.
Prior Knowledge & Skills
 Use of a graphing calculator to organize data in lists, compute measures of one-variable statistics,
and construct a histogram.
Planning Notes
 Students require a graphing calculator or access to a spreadsheet or statistical software (e.g., Fathom)
and Internet access. Students can download the data for income by family from E-Stat. From the
section entitled People on the Table of Contents, they select Personal Finance and Household
Finance (click on Data at the bottom of the page, then Income under Cansim II). It is suggested
that the income from Table 202-0401 be used because the income intervals go up to $150 000. The
individual income (Table 202-0101) data only goes up to income of $60 000, but may be interesting
data for further study of this issue. The CPI data can also be downloaded from E-Stat, under
Economy, Prices and Price Indexes. From the list of indexes, choose Consumer Price Index.
(Table 326-0002).
Unit 5 – Page 6
 Mathematics of Data Management – University Preparation

Students work individually or in pairs. The pairing could be done randomly or by ability. It may be
appropriate to pair students of similar ability levels, but it may also be effective to pair strong and
weak students.
Teaching/Learning Strategies
The purpose of this activity is to introduce students to a significant problem: “Is the gap between the rich
and poor in Canada widening?” Students should discuss data or information that would be useful in
answering this question. This could be a sensitive topic for some students. A search of Internet sites
using the key words “gap, rich, poor” will result in numerous articles and data which can be used as an
introduction as well as provide opportunities for further study. One article, which examines the gap in
Canada and the US, can be found at www.statcan.ca/Daily/English/000728/d000728a.html
Ways to measure the gap between rich and poor may include: the number of millionaires, number of
people who live below the poverty line, range of income, measures of spread, etc. An article from the
University of Toronto Varsity News, July 24, 2001 entitled “Youth hit hard as gap between rich and poor
grows, says report” may facilitate discussion
(www.varsity.utoronto.ca/archives/119/oct26/news/youth.html).
If Internet access is limited, the data could be provided to students. (See Appendix 5.2.1.) Students need
to discuss ways to analyse and describe characteristics of the data. Due to the volume of the data,
suggestions can be made to look at specific years at regular intervals (e.g., when considering data from
1980 to 1998, it may be useful to consider data from regular time periods – for example: 1980, 1986,
1992, and 1998) To reduce the number of intervals, a suggestion may be to use $25 000 intervals. The
midpoints of the intervals are used when calculating the measures of central tendency.
Questions to consider:
 What effect might reducing the number of intervals have on the results?
 What are the implications of using $12 500 as the midpoint of the first income interval?
 Is it reasonable to use $162 500 as the midpoint of the over $150 000 interval?
The data is entered into a graphing calculator or computer software. The midpoints of the intervals are
placed in L1 and the percentage of earners in each income interval placed in L2 through L5 for each of the
four years considered. The mean, median, standard deviation, and quartiles can be calculated for each
year using the one variable statistics on L1 and each of the four other lists. Compare these measures for
each of the four years.
Questions to consider:
 Are your calculations the same as those provided by Statistics Canada? What might account for the
differences?
 Are there any patterns emerging?
 How have each of the measures changed from year to year?
 What do the changes in each measure tell us about the data?
 What is the percentage change in each of the measures from year to year?
 Which measures show the greatest percentage change for which years?
 What does the change tell us about the income levels of Canadians over these years?
 When the z-score is calculated for a family with an income of $30 000, how has the z-score changed
over the years?
 If you repeat the calculations for an income of $100 000, what conclusions can be made?
Frequency distributions for individual years can be drawn using the graphing calculator. Different
distributions can be drawn by hand and placed on one graph using four different colours so that the
distributions can be compared.
Unit 5 – Page 7
 Mathematics of Data Management – University Preparation



What similarities and differences do you notice in the distributions?
Are any patterns apparent?
What do these patterns tell us about the family income trends in Canada from 1980 to 1998?
Calculate the changes in percentages at each income level from 1980-1998 and construct a line graph of
these changes. Examine the changes in the frequencies for each income level.
 What changes are evident?
 What does this tell us about the trends in income?
 Does this help us answer the big question?
Use the Consumer Price Index to adjust the figures in terms of 1998 dollars. This can be done by
multiplying the income interval midpoints by a factor of 108.6/108.6 = 1 for 1998, 100/108.6 = .9208 for
1992, 78.1/108.6 = 0.7192 for 1986 and 52.4/108.6 = 0.4825 for 1980.
Recalculate the mean, median, standard deviation, and quartiles for these inflation-adjusted figures.
Organize the results in a new table.
Questions to explore:
 Do these adjusted figures change any of your previous conclusions?
 What factors may have caused family income in Canada to fall well behind inflation rates?
 How has the Canadian family changed from 1980 to 1998 (e.g., income earners, double-income
families, adult children contributing to family income, divorces, single-parent families, part-time
employment, etc.)?
Some students could be encouraged to pursue this issue further in a culminating project.
Assessment & Evaluation of Student Achievement
 The purpose of this activity is to provide students with the opportunity to use skills that will be
necessary in their culminating project. Each student should have an opportunity during the course to
prepare a written report, make a presentation, and critique the work of other students before they
apply these skills to the culminating project.
 At this time it would be reasonable for some students (or some pairs of students) to submit a written
report. Preparation of the report would require the use of graphing software to print graphs of their
data. Exposure to the technology is important so that students feel comfortable using the software in
their culminating projects. The written report should be assessed for mathematical content. (See
sample rubric in Appendix B.)
Accommodations
Students’ IEPs may suggest ways to support student learning that would be appropriate for this activity.
Extending an activity into a culminating project is a way for students with special learning needs to get a
“jumpstart” on their projects.
Resources
http://estat.statcan.ca Income tables 202-0401,202-0101, CPI table 326-0002
www.ontario.cmha.ca Canadian Mental Health Association, Backgrounder on Poverty, November 2000
This report from the Ontario Child Health Study: Children at Risk examines the correlation between
being poor and having a much greater risk of suffering from mental health problems.
www.statcan.ca/Daily/English/00728/d00728a.htm - Article examines income inequality (gap between
the rich and poor) in Canada and U.S.
www.varsity.utoronto.ca/archives/119/oct26/news/youth.html - Report states “Youth hit hard as gap
between rich and poor grows.”
Unit 5 – Page 8
 Mathematics of Data Management – University Preparation
Appendix 5.2.1
Student Datasheet
Group
L1980
“Avg inc”
47703
“Med inc”
42250
0-5000
2.8
5000-9999
5.5
10000-14999
8
15000-19999
6.9
20000-24999
6.2
25000-29999
5.7
30000-34999
6
35000-39999
6
40000-44999
6.1
45000-49999
5.7
50000-54999
6.1
55000-59999
5.4
60000-64999
4.6
65000-69999
4.2
70000-74999
3.7
75000-79999
3.1
80000-84999
2.6
85000-89999
2
90000-99999
2.8
100000-124999
3.8
125000-149999
1.5
150000 and over 1.4
Unit 5 – Page 9
L1981
47479
41716
2.2
5.3
8.3
6.6
6.5
6.3
6.3
6.4
6.1
5.8
5.8
5.2
4.5
4.5
3.5
2.6
2.5
2.2
3
3.9
1.5
1.1
L1982
46563
40416
2.3
5.1
8.5
6.8
7.2
6.8
6.2
6.5
6.3
6
5.4
5.1
4.4
3.9
3.3
2.8
2.2
1.9
2.9
3.6
1.5
1.2
L1983
45451
38895
2.6
5.9
9.3
7
7.6
6.5
6.3
6.4
6.2
5.7
4.8
5.2
4.2
3.5
3.1
2.7
2.3
1.9
2.8
3.3
1.6
1.2
L1984
45608
39216
2.5
5.3
9.2
7.3
7.7
6.6
6.3
6.1
5.9
5.9
5.7
4.9
4.3
3.7
3.2
2.7
2.1
1.8
2.7
3.5
1.3
1.3
L1985
46526
39613
2.3
4.9
9.1
7.3
7.6
6.7
6.4
6.2
5.9
5.8
5.3
5
4.2
3.8
3.4
2.7
2.3
2
2.8
3.8
1.5
1.2
L1986
47106
40062
2.2
4.5
9.2
7.6
7.4
6.7
6.5
5.9
6.1
5.6
5.2
5
4.1
3.9
3.4
2.8
2.4
2
2.9
3.8
1.5
1.5
 Mathematics of Data Management – University Preparation
Appendix 5.2.1 (Continued)
L1987
47420
39975
2
5
9
7.1
7.7
6.9
6.2
6.3
5.8
5.8
5.3
4.7
4.1
3.8
3.3
2.6
2.3
2
3
4
1.6
1.6
L1988
48429
4730
1.8
4.8
8.9
7
7.3
6.8
6.4
6.1
5.6
5.7
5.2
4.5
4.2
4.1
3.2
2.9
2.4
2.1
3.2
4.4
1.6
1.8
L1989
49913
42455
1.5
4
8.5
7.2
7.3
6.2
6.3
6.3
5.9
5.9
5.3
4.9
4.4
3.7
3.2
3.1
2.5
2.1
3.3
4.7
1.9
1.8
L1990
49116
41294
1.7
4.2
8.8
7.2
7.6
6.6
6.2
6.1
5.6
5.7
5
4.8
4.4
3.8
3.3
2.8
2.6
2.1
3.4
4.5
1.7
1.9
L1991
47487
39137
2
4.9
8.9
7.6
7.5
6.9
6.9
6.5
5.6
5.5
4.6
4.8
3.9
3.6
3.5
2.7
2.2
1.9
3.1
4.1
1.7
1.6
L1992
47603
39766
2
4.7
8.9
7.7
7.4
7.1
6.7
5.8
6.1
5.3
5
4.6
4.1
3.7
3.2
2.8
2.2
2.1
3.3
4.3
1.6
1.6
L1993
46416
38061
2.1
4.7
9.4
8.1
8
6.8
7.1
6
5.9
5.1
4.7
4.3
4.4
3.2
3.1
2.6
2.3
2
3.1
3.9
1.5
1.4
Year
1980
1981
1982
1983
1984
1985
1986
1987
1988
1989
1990
1991
1992
1993
1994
1995
1996
1997
1998
Unit 5 – Page 10
L1994
47254
39255
1.7
5.2
8.8
8.6
7.4
6.5
6.4
6.3
5.9
5.1
5.2
4.6
4.3
3.6
3.1
2.5
2.2
1.9
3
4.3
1.9
1.5
L1995
47246
38608
1.9
4.8
8.9
8
8.1
6.7
6.9
6.3
6
5.2
4.8
4.4
3.9
3.6
3
2.7
2.3
2
2.9
4
1.6
1.7
L1996
47476
38613
2.4
5.2
9.4
7.8
7.4
7.1
6.4
5.9
5.8
4.8
4.9
4.6
4.2
3.7
3.1
2.7
2.4
1.9
2.7
4.4
1.7
1.7
L1997
48124
38325
2.4
5.2
9.3
7.8
7.4
7
6.7
6.2
5.2
4.9
4.8
4.2
4.2
3.6
3
2.6
2.3
1.8
3.3
4.1
1.9
2
L1998
49797
39398
2.3
5
9.1
7.3
7.3
7.2
6.7
5.8
5.5
4.9
4.7
4.5
3.6
3.7
3.3
2.8
2.1
2
3.2
4.9
2
2.3
CP1
52.4
58.9
65.3
69.1
72.1
75
78.1
81.5
84.8
89
93.3
98.5
100
101.8
102
104.2
105.9
107.6
108.6
 Mathematics of Data Management – University Preparation
Activity 5.3: AIDS in Canada – A Modelling and Simulation Activity
Time: 4 hours [use suggested at or near the end of Unit 2]
Description
Students analyse data relating to the spread of AIDS in Canada over the last twenty years. Measures of
central tendency are used to describe the characteristics of the data and their applicability as predictors.
The trends in the data over time are examined, and a linear model applied. The correlation coefficient
and the graph are used to consider the appropriateness of the linear model in making predictions. A
simulation is constructed and the results compared and contrasted with the actual data. Adjustments are
made to the simulation model until the produced data closely reflects the actual data. Once refined, the
data from the simulation is used to make predictions about the future. The reliability of this model is
assessed.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE1d - develops attitudes and values founded on Catholic social teaching and acts to promote social
responsibility, human solidarity, and the common good;
CGE2a - listens actively and critically to understand and learn in light of gospel values.
Overall Expectations
ODV.01 - organize data to facilitate manipulation and retrieval;
CPV.03 - design and carry out simulations to estimate probabilities;
STV.01 - demonstrate an understanding of standard techniques for collecting data;
STV.04 - describe the relationship between two variables by interpreting the correlation coefficient.
Specific Expectations
OD1.02 - use the Internet effectively as a source for databases;
CP3.01 - identify the advantages of using simulations in contexts;
CP3.02 - design and carry out simulations to estimate probabilities;
ST1.04 - organize and summarize data from secondary sources using technology;
ST4.01 - define the correlation coefficient as a measure of the fit of a scatter graph to a linear model;
ST4.02 - calculate the correlation coefficient for a set of data, using graphing calculators or statistical
software;
ST4.04 - describe possible misuses of regression.
Prior Knowledge & Skills
Use of a graphing calculator to organize data in lists, compute measures of one-variable statistics,
construct a scatter plot, perform a linear regression, use formulas in lists, and generate random numbers.
Planning Notes
The data can be provided or students can access the data at www.hc-sc.gc.ca. Students work in pairs.
Each pair requires a graphing calculator. Students should be given the worksheet to facilitate the
collection of the data during the simulation. Access to software for printing graphs may be useful (e.g.,
TI-Graphlink, Fathom, TI-Interactive, Excel, or Quattro Pro).
Teaching/Learning Strategies
The teacher should be sensitive to individual circumstances. The main questions are “What is the
predicted future growth of AIDS cases in Canada?” and “Can we use past data to predict what might
happen when there is an outbreak of a disease?”
Unit 5 – Page 11
 Mathematics of Data Management – University Preparation
In the early 1990s, there was a fear that AIDS would become an epidemic and it would run rampant
throughout the population. Discussion of factors that might influence the spread of AIDS could include
age, attitude about risk, education, and prevention.
The table gives the number of reported AIDS cases by year of diagnosis:
Year
1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989
# of Cases
1
5
9
26
66
164
375
632
953
1159 1387
1990
1433
1991
1556
1992
1732
1993
1758
1994
1733
1995
1579
1996
1063
1997
688
1998
599
1999
415
2000
261
After students examine the data, discuss underlying reasons for the trends that are evident.
Students enter the number of cases from 1979 to 1998, a span of twenty years, into a list on a graphing
calculator. Enter the years into L1 using 1979 as year 1 and the number of cases into L2. (Retain the data
for subsequent years as a test of the validity of the models developed.)
Method 1: Measures of Central Tendency
Students calculate the measures of central tendency using the graphing calculator and use them to
describe the characteristics of this set of data. The mean, median, or mode is often useful to represent a
set of data. Does the mean or median appear to be a good predictor for this data? Why would the mode
not be used? Compare the prediction with the actual figures for the year 2000. What if only the last ten
years were used? Would you use the mean or median as a predictor? Justify.
Method 2: Line of Best Fit
In order to investigate the trend over time, students consider the cumulative number of cases reported.
The cumulative sum can be entered into L3 using the cumSum command in the List OPS menu of the
graphing calculator. Next, students consider the relationship between two variables by looking at the
trends in the data over time. Create a scatter plot of the total cases reported versus the year (1-20). Use a
linear regression to determine the line of best fit and consider the resulting correlation coefficient.
Students should assess the validity of the model on the basis of this coefficient and the graph. Does it
make sense that the number infected per year increases and then decreases? Use the linear model to
predict the number of reported cases of AIDS in the year 2000. Compare the prediction to the actual
number of reported cases. Is the prediction accurate?
Further questions to consider:
 In theory, if we extrapolate with the linear model, the whole population will be infected. Is this
realistic?
 Are there any outliers? What if we excluded the outliers?
 What if we used only the last ten years? Is this model a good fit?
 What are some limitations of the linear model?
Method 3: Simulation
The simulation models the spread of an infectious disease in a population. Although the original design is
laid out, there is opportunity for students to redesign the model.
Suppose there are 100 people in a population and one of these people is infected with a disease. Suppose
each infected person infects one other person in the population each year. Using a graphing calculator,
students use random numbers between 1 and 100 to model the spread of the infection. Construct a table
containing numbers 1 to 100 to keep track of those infected over a 20-year period. At the start (year 0),
one person is infected. Use the random number generator on a graphing calculator to select the infected
person. Place an X in the chart to indicate infection. In year 1, another person becomes infected (chosen
by a random number). In year 2, both infected people come into contact with other people (chosen by two
random numbers) and those people become infected. Indicate these with X’s. The random integer
Unit 5 – Page 12
 Mathematics of Data Management – University Preparation
function can be used to produce a specific number of random numbers (e.g., if there are 12 infected
people in a particular year, during the following year, 12 more people could become infected). By using
RandInt (1,100,12), 12 random numbers appear on the screen. You may need to scroll to see all the
numbers.
In a second table, keep track of the cumulative number infected in the population for each year.
Remind students that, once infected, a person cannot be re-infected, so some of the contacts will not
result in new infections. Continue until year 20 (or when the entire population is infected).
Students who are familiar with computer programming may design a computer program to run this
simulation (see Turing and C++ examples in Appendix E). A TI-83 program [available at
www.ugdsb.on.ca\cddhs\math] could be used.
Doing a simulation only once is not sufficient to conclude that this is the pattern. Why might this be the
case? Often a simulation needs to be repeated a number of times to see if a pattern develops. Use a table
to collect cumulative number of infections from other students’ simulations and average the results. Is
there a pattern developing? On average, how many years did it take for the entire population to be
infected?
Place the simulated number of cases (averaged from the simulations done by the class) in L4. Graph the
simulated data L4 versus L1. Compare this graph to the graph of the AIDS data (L3 versus L1). How does
the simulation compare as a model?
Students should notice that the general shape of the graph of the simulated data is similar to the actual
data; there is a sharp increase in the number of cases initially and then a levelling off as time progresses.
The simulation model is certainly better than the linear model - in general the simulation curve appears to
be similar to the actual. It may increase or decrease too quickly and, of course, the entire population is
infected in the simulation.
Is it possible to map the simulation data so that it is similar to the actual curve? Since Canada has a
population of about 30 million and our population was 100, we could multiply the cases by 300 000.
Does this appear realistic? Explain. The simulation was done with a population of 100, with all persons
having the same likelihood of contracting the disease. Is this realistic? Explain. The rate of increase may
be too quick or slow in our simulation. We assumed an infection every year. It may be less or more.
Multiply your time values (L1) by an appropriate factor. Place this data in L5. (Try 0.5 or 2 to get a sense
of how the graph might change.)
The number infected in our population in comparison is quite different. Multiplying by 300 000 is not
realistic. Multiply the infected column (L4) by a factor. Place this data in L6. (Try 50 or 100 to get a sense
of how the graph might change.)
Using the modified simulation model, what would you predict about future trends for AIDS cases in
Canada? Is this an appropriate model? Justify.
Some further questions to consider:
 Why does this levelling out in the graphs occur?
 What are some of the factors affecting the maximum number of people that are likely to become
infected over time?
 What are some of the limitations of this simulated model?
Research the incidence of AIDS in other countries. Do the same trends appear?
Research the incidence of other diseases within populations. Are the trends similar?
Once students have completed the activity, the class could discuss: What information should be included
in a written report and how should it be displayed? What should be included in a presentation? What
further investigation is appropriate?
Unit 5 – Page 13
 Mathematics of Data Management – University Preparation
Assessment & Evaluation of Student Achievement
 The purpose of this activity is to provide students with the opportunity to use skills that will be
necessary in their culminating project. Some pairs of students may submit a written report, which
should be assessed for mathematical content. (See rubric in Appendix B)
 Some pairs may make a presentation to the class. Teachers assess students on their presentation skills
as well as the mathematical content of their presentation. (See rubric in Appendix C.) Refer to
Activity 5.6 for more suggestions about presentations and their assessment.
 Students should practise critiquing the presentations of other students for effectiveness. Feedback
from peers should not be part of the student’s assessment. Critiquing is intended to provide feedback
(positive as much as possible); feedback should be viewed as formative assessment to help students
improve their presentation skills. The presentation rubric in Appendix C could be used by teachers
and students, or a simpler checklist, such as the sample in Appendix D, might be easier for students
with little experience with critiquing.) Refer to Activity 5.6.
Resources
Health Canada – www.hc-sc.gc.ca (data on Aids cases)
Activity 5.4: Dice Differences and the Non-Transitivity Paradox
Time: 3 hours
Description
Dice games have been popular since ancient times. The study of probability began because of interest in
games of dice. The famous French nobleman and professional gambler Chevalier de Mere (1607-1684)
corresponded with Blaise Pascal about his chances in a dice game. De Mere wanted to know which had
4
5
the higher probability: getting at least one “6” in four rolls of a die ( 1     0.5177(approx.) ) ) or
6
24
 35 
getting at least one double-six in 24 throws of two dice ( 1     0.4914(approx.) ) ) . De Mere
 36 
suspected that the first had a higher probability than the second, but his mathematical skills were not
great enough to demonstrate why this should be so. De Mere’s observation remains true even if two dice
are thrown 25 times, since the probability of throwing at least one double-six is then
25
 35 
1     0.5.55(approx.) .. This dice problem has since been known as de Mere’s Problem. A
 36 
National Film Board Video, Of Dice and Men, goes through the history of De Mere and introduces
probability.
The challenge is that the probability is not always what it seems in many games. Often the appearance of
fairness has been used to the advantage of others in gambling situations. In this activity, students
consider two dice games: one involves dice differences and the other involves non-transitive dice. (The
non-transitivity paradox is where although A is preferred to B and B is preferred to C, A is not preferred
to C.) Students examine the probability concepts that underlie both games.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE5a - works effectively as an interdependent team member;
CGE3b - creates, adapts, and evaluates new ideas in light of the common good.
Unit 5 – Page 14
 Mathematics of Data Management – University Preparation
Overall Expectations
CPV.02 - determine and interpret theoretical probabilities, using combinatorial techniques;
CPV.03 - design and carry out simulations to estimate probabilities.
Specific Expectations
CP2.01 - solve probability problems involving combinations of simple events, using counting techniques;
CP2.02 - identify examples of discrete random variables;
CP2.04 - calculate expected values and interpret them within applications as averages over a large
number of trials;
CP2.05 - determine probabilities using the binomial distribution;
CP2.06 - interpret probability statements, including statements about odds, from a variety of sources;
CP3.01 - identify the advantages of using simulations in contexts;
CP3.02 - design and carry out simulations to estimate probabilities;
CP3.03 - assess the validity of some simulation results by comparing them with the theoretical
probabilities.
Prior Knowledge & Skills
Students should understand the basic concepts of probability. The experimental part of each activity can
be used as an introduction to the unit on probability. Students can return to the activity and investigate
the underlying theoretical probabilities after they have studied the concepts in more detail.
Planning Notes
Note: This activity uses data from the use of dice. Teachers should handle the activity in such a way to
ensure that the gambling aspect is not glamorized or presented as a positive adventure.
Part A - Dice Differences
Students play the game in pairs. Each set of players needs a pair of dice or a TI-83 graphing calculator
with the DICEDIFF program.
Part B: Non-Transitivity Paradox
Non-transitive dice demonstrate a probability that challenges our intuition and traps the unwary. You
may want to introduce the topic by telling students about a meeting between Warren Buffet, the investor
and Bill Gates, the chairman of Microsoft. In an article by Andrew Kupfer in Fortune Magazine
dated 02-05-1996, Bill Gates describes his relationship with Buffet:
“One area in which we do joust now and then is mathematics. Once Warren presented me with four
unusual dice, each with a unique combination of numbers (from 0 to 12) on its faces. He proposed
that we each choose one of the dice, discard the third and fourth and wager who would roll the
highest number most often. He graciously offered to let me choose first. Then he said, “Okay,
because you get to pick first, what kind of odds will you give me?” I knew something was up. “Let
me look at those dice”, I said. After studying the numbers on their faces for a moment, I said, “This is
a losing proposition. You choose first.” Once he chose a die, it took me a couple of minutes to figure
out which of the three remaining dice to choose in response. Because of the careful selection of the
numbers on each die, they were non-transitive. Each of the four dice could be beaten by one of the
others: die A would tend to beat die B, die B would tend to beat die C, die C would tend to beat die
D, and die D would tend to beat die A. This meant that there was no winning first choice of a die,
only a winning second choice. It was counterintuitive, like a lot of things in the business world.”
It may be most effective to construct a large set of non-transitive dice to be used in the classroom setting.
Challenge students to pick one die in a game of The Best of Ten Throws. By choosing the appropriate
die, odds are such that you should win almost every time. There are several sets of non-transitive dice.
(They can be purchased online – www.grand-illusions.com/magicdice.htm)
Unit 5 – Page 15
 Mathematics of Data Management – University Preparation
One possible set is:
Dice A: 6,6,2,2,2,2
Dice B: 5,5,5,1,1,1
Dice C: 4,4,4,4,0,0
Dice D: 3,3,3,3,3,3
(Other sets may have an increased probability of winning, but with this set the probability between each
pair of dice is the same.)
www.grand-illusions.com/magicdice.htm
Other similar sets are:
Dice A: 11,10,9,3,3,2
Dice B: 8,8,8,7,1,0
Dice C: 6,6,6,6,5,5
Dice D: 12,12,4,4,4,4
Some three dice non-transitive sets are:
Dice A: 4,4,4,4,1,1
Dice B: 3,3,3,3,3,3
Dice C: 5,5,2,2,2,2
Dice A: 11,10,9,3,2,1
Dice B: 9,8,8,7,1,0
Dice C: 7,7,6,6,5,5
Dice D: 12,11,5,4,4,3
Dice A: 7,7,5,5,3,3
Dice B: 9,9,4,4,2,2
Dice C: 8,8,6,6,1,1
Teaching/Learning Strategies
Part A - Dice Differences
One player is the Low player and the other is the High player. The dice are rolled and the difference is
calculated. The differences will range from zero (if the upfaces are the same) to 5 (if a 1 and a 6 are
rolled). If the dice difference is 0, 1, or 2, the Low player wins. If the dice difference is 3, 4, or 5, the
High player wins. Students play 25 rounds and record who wins each round and who wins overall.
Note: Students can get the necessary data quickly using a TI-83 program instead of rolling the dice (see
the TI-83 manual for more detailed instructions):
PROGRAM:DICEDIFF
:Lbl D
:rand Int(1,6) > A
:rand Int(1,6) > B
:Disp abs(A-B)
:Pause
:Goto D
Once the program is executed, the student presses the Enter key seven times to get a new screen with
seven new pieces of data, recording them in a tally chart, seven at a time.
Does the game appear to be fair to both players?
Unit 5 – Page 16
 Mathematics of Data Management – University Preparation
Data from the whole class can be compiled. Use the class results to determine the experimental
probability of each player winning. Compute the theoretical probability of each difference outcome. (A
grid of the possible outcomes could be suggested to students as a strategy.) Construct a probability
distribution for the possible outcomes.
Compute the theoretical probability of each player winning.
It turns out that the game is not fair. How could the rules be changed to make it fair? Compute the
probabilities for the altered game to illustrate the fairness.
A variation of this game is Prisoners. Each player requires a playing board of 6 cells, 6 counters to
represent the prisoners, and 2 dice (for each pair). Each player places their prisoners into any cells on
their own game board. Players may place one prisoner in each cell, or two in some cells and none in
others, or all six in one cell. Take turns rolling the dice. Calculate the difference. The player rolling the
dice may release one prisoner from the cell with that number. The winner is the first to release all their
prisoners.
Keep a record of where you place your prisoners for each game, then record the ones that were winners.
What combination appears to be a winning strategy? Compare your results with the rest of the class.
Cell
Winners
0
1
2
3
4
5
2
2
1
1
0
0
X
Test two of the best combinations from the class against each other. Record the number of tosses it takes
to release all the prisoners for each combination, as well as recording the winner. (If one combination
wins after 8 tosses, keep tossing until all prisoners are released for the losing combination as well.)
Repeat the game several times. Compile the results for the class.
Tally the experimental results in a table for each of the combinations (e.g., if Combination 2 won the
game in 9 tosses and Combination 1 took 11 tosses to release all prisoners, tally the results):
Number of Rolls
Combination 1
Comb. 1 Wins
Combination 2
Comb. 2 Wins
Necessary
6
7
8
9
X
X
10
11
X
12
13
14
15
More than 15
Totals
Compute the experimental probability that all prisoners will be released in 6 tosses, 7 tosses, etc. for each
of the combinations. What is the experimental probability that one combination will win over the other
combination? Compute the theoretical probability that all prisoners will be released in 6 tosses, 7 tosses,
etc., for each of the combinations. Do the experimental probabilities approximate these theoretical
probabilities? Which combination should be the most successful in theory? Do the experimental results
support this theory?
Unit 5 – Page 17
 Mathematics of Data Management – University Preparation
Part B: Non-Transitivity Paradox
As the class challenge is carried out, students record the winner of each roll as well as the winner in The
Best of 10 Throws. What is the experimental probability that the teacher will win on a single roll? What
is the theoretical probability that die A beats B? Students being introduced to probability may use a grid:
The Winning Player
A/B
5
5
5
1
1
1
6
A
A
A
A
A
A
6
A
A
A
A
A
A
2
B
B
B
A
A
A
2
B
B
B
A
A
A
2
B
B
B
A
A
A
2
B
B
B
A
A
A
Similarly, calculate the probability that B beats C, C beats D, and D beats A. What is the probability that
the teacher wins in exactly 5 games? What is the probability that the teacher wins in exactly 6 games?
7 games? 8 games? 9 games? 10 games?
Individual Calculations for Winning in 5 Games and 10 Games (http://exploringdata.cqu.edu.au)
Teacher
5 Games
Student
No. games
5
P(wins game)
0.667
Probability teacher wins in:
5
0.132
6
0.219
7
0.219
8
0.171
9
0.114
Total
0.855
No. games
5
P(wins game)
0.333
Probability student wins in:
5
0.004
6
0.014
7
0.027
8
0.043
9
0.057
Total
0.145
10 Games
Teacher
Student
No. games
10
No. games
10
P(wins game)
0.667
P(wins game)
0.333
Probability teacher wins in:
Probability student wins in:
10
0.017
10
0.0000
11
0.058
11
0.0001
12
0.106
12
0.0004
13
0.141
13
0.0011
14
0.153
14
0.0024
15
0.143
15
0.0045
16
0.119
16
0.0074
17
0.091
17
0.0113
18
0.064
18
0.0161
19
0.043
19
0.0214
Total
0.935
Total
0.0648
Design a simulation to model the results with this set of dice. How do the simulated results compare with
the theoretical probabilities? Investigate other sets of non-transitive dice. Try creating your own set.
Compute the resulting probabilities. Test your set and compare the experimental results with the
theoretical probabilities of winning. Design a simulation to model the results of this set of dice.
Have a class discussion about how these probability concepts might apply to the culminating projects.
Unit 5 – Page 18
 Mathematics of Data Management – University Preparation
Assessment & Evaluation of Student Achievement
Part A - Dice Differences
 Students could present the dice differences game with the altered rules to the class. If time allows,
the class could play the altered game and determine if the results illustrated the theoretical
probabilities that students had determined. The open-ended nature of designing their own rules may
lead students towards further investigation in game theory.
 Prisoners can be used as a class activity. The emphasis on experimental results is handled more
easily by compiling class results. It can be used to introduce the concept of probability distributions,
or as an assessment after students have learned the concepts. For assessment, students might work in
pairs and present their findings, both experimental and theoretical, in the form of a written report.
Part B: Non-Transitivity Paradox
 This activity can be used as an introduction to the probability unit. Experimental results can be
collected. Then, after the probability concepts have been fully developed, students can return to the
unit and complete the theoretical computations and design the simulations.
 If used as a summary activity for the unit on probability, students could work in pairs to collect the
experimental data. A written report comparing the experimental results with the theoretical
computations could be part of the assessment.
 If students construct their own set of non-transitive dice or design a simulation of a set of dice, a
class presentation may be appropriate.
Accommodations
Students with similar skill levels should be paired. This allows more time for the teacher to assist
students experiencing difficulty. See the overview for further suggestions for accommodations.
Resources
http://exploringdata.cqu.edu.au/ A resource for data and other probability activities.
National Film Board. Of Dice and Men. A video about De Mere and probability concepts.
www.grand-illusions.com/magicdice.htm A website offering non-transitive dice (magic dice)
Activity 5.5: Using Unit 4 Tools
Time: 1 hour
Description
The tools from Unit 4 may provide students with an alternative way to organize and present their data.
This activity considers tools that may be useful in some culminating projects and suggest possible
avenues for students to consider.
One suggestion for a culminating project may be a study of fractals. Fractals are self-replicating shapes
(i.e., as you focus on a portion of the shape it maintains the same geometrical shape). This part of the
activity looks at one of the most famous fractals, Sierpinski's Triangle.
Another focus for a culminating project may be on problems involving networking, graph theory, and
matrices. A networking problem may be illustrated in the form of a graph using vertices and edges or a
matrix where each row and column represents a vertex in the network. This part of the activity looks at
another famous problem - the Koenigsberg bridge problem.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE2b - reads, understands, and uses written materials effectively;
CGE3b - creates, adapts, and evaluates new ideas in light of the common good.
Unit 5 – Page 19
 Mathematics of Data Management – University Preparation
Overall Expectations
ODV.02 - solve problems involving complex relationships, with the aid of diagrams;
ODV.03 - model situations and solve problems involving large amounts of information using matrices.
Specific Expectations
OD2.01 - represent simple iterative processes using diagrams that involve branches and loops;
OD2.02 - represent complex tasks or issues using diagrams;
OD2.03 - solve network problems using introductory graph theory;
OD3.01 - represent numerical data, using matrices and demonstrate an understanding of terminology and
notation related to matrices;
OD3.02 - demonstrate proficiency in matrix operations, including addition, scalar multiplication, matrix
multiplication, the calculation of row sums and the calculation of column sums, as necessary to solve
problems, with and without the aid of technology;
OD3.03 - solve problems drawn from a variety of applications using matrix methods.
Prior Knowledge & Skills
These activities may be used to introduce the concepts of iterative processes and graph theory or as
samples at the end of the unit to prompt further exploration.
Planning Notes
Each part of the activity is independent. Students may look at just one of the components of iteration,
networking, or matrices.
Teaching/Learning Strategies
Part A: The Sierpinski Triangle:
One of the most famous fractals is Sierpinski's Triangle.
An algorithm for it is as follows:
1. Draw a triangle.
2. Construct a point at the midpoint of each side and form another triangle using the three midpoints as
vertices. Four triangles result: one in the middle, and three surrounding it.
3. Shade in the newly formed middle triangle.
4. Repeat steps 2 and 3 for the three white triangles surrounding the newly created one as often as
possible.
Perform the above algorithm to draw Sierpinski’s Triangle.
Geometer’s Sketchpad contains a pre-made script that creates Sierpinski’s triangle. The file, sirpnski.gss,
can be located under \samples\scripts\fractals\sirpnski.
It is possible to generate the triangle by introducing randomness. Although points are drawn based on the
selection of a random number, the same image results after many iterations. It is possible to construct by
hand but many iterations are required. Results are better if a computer program is used that can quickly
perform the iterations. The Guide Book for the TI-83 Plus calculator has a sample program on pp. 17-7.
The algorithm is as follows:
1. Set up a grid of a reasonable size, a suggestion may be 12 by 12.
2. Randomly select starting coordinates x and y.
3. Roll a die.
4. If 1 or 2 is rolled, recalculate x and y on the basis of the following: x = 0.5x, y = 0.5y
5. If 3 or 4 is rolled, recalculate x and y on the basis of the following: x = 0.5(0.5+x), y = 0.5(1+y)
6. If 5 or 6 is rolled, recalculate x and y on the basis of the following: x = 0.5(1+x), y = 0.5y
7. Plot the new point.
8. Repeat steps 3 to 7.
Perform the algorithm a number of times to draw Sierpinski's Triangle.
Unit 5 – Page 20
 Mathematics of Data Management – University Preparation
Extensions: Research other fractals and describe the steps in an algorithm or in a flow chart (e.g., Koch’s
Snowflake, Julia Sets, Barnsley’s Fern, Mandelbrot set, etc.). If you are proficient in computer
programming or programming on the calculator you may create your own program. Michael Barnsley,
author of Fractals Everywhere, suggests it is possible to determine a fractal algorithm for a given image
and send the algorithm over the Internet rather than the actual image, thus reducing the time taken for
transferring information.
Part B: The Seven Bridges of Konigsberg
Konigsberg was a city in Prussia situated on the Pregel River. The city is called Kaliningrad today and is
a major industrial centre in western Russia. The river Pregel flowed through the town creating an island.
Seven bridges spanned the various branches of the river. Some of the town’s curious citizens wondered if
it was possible to travel across all seven bridges without having to cross any bridge more than once. All
who tried, failed.
The problem can also be described as drawing the picture without retracing any line and without lifting
the pencil from the paper. If the path is traceable, it is called an Euler path. Students might investigate
various diagrams, identifying Euler paths or create challenge diagrams for classmates to try.
www.contracosta.cc.ca.us/math/Konig
www.jcu.edu/math/vignette/bridges.htm
Network problems such as this can be solved using graph theory. Name each of the land areas as the
vertices and the bridges joining these areas as the edges. The order of a vertex is the number of edges at
that vertex. If we consider one of the vertices with order 3: If we start at this vertex, we leave by one
bridge, and return by another. This means that we must leave again to cross the third bridge. If we start
elsewhere, we would pass through this vertex by entering on one bridge, leaving by another. We would
have to return to cross the third bridge, therefore ending at that vertex. This means our trip must either
begin or end on this vertex since it has three edges, making it an odd vertex. The same argument would
hold for the other vertices, which are also odd. Since we cannot begin or end at more than two vertices,
the trip is impossible. (To be possible, there would have to be just two odd vertices. We would begin at
one and end at the other.)
Such problems can be put in a context. (Suppose you are given the map of a town and told to design a
route for snowplows. Is it possible to plough every street without ploughing any street more than once?)
Students are encouraged to create a context (e.g., see www.jcu.edu/math/vignettes/bridges,
www.bridges.canterbury.ac.nz/features/bridges, www.sunybroome.edu/~mat_dept/current/113konig,
www.contracosta.cc.ca.us/math/Konig).
Network problems can be modelled using matrices. Each row or column would represent the vertex and
the entries of the matrix represent the number of direct edges joining the vertices.
If we consider the land masses A, B, C, and D, the network matrix can be formed:
A
A 0
B  2
T
C 2

D 1
B
2
0
0
1
Unit 5 – Page 21
C
2
0
0
1
D
1
9

1  2 1
T 
1
1


0
4
1 1 4
5 5 2 
5 5 2

2 2 3
 Mathematics of Data Management – University Preparation
The network matrix T and powers of that matrix allow us to consider other questions:
How many ways can you get from A to B by crossing one bridge?
How many ways can you get from A to B by crossing two bridges?
Is it possible to get from B to C without crossing at least two bridges?
Matrices are useful when considering large networking problems. Manipulation of the matrices can be
done with technology. Students may research the matrix applications in networking problems.
Have a class discussion about how the skills and concepts in this activity could be applied to the
culminating project. Suggest Markov chains, communication matrices, profit/cost matrices, etc. to show
students how they can use matrices to connect to their projects.
Assessment & Evaluation of Student Achievement
 Students who work on these activities might prepare a written report and/or make a presentation to
the class. Students could critique the presentations. (See rubrics in Appendix B, C, and D.)
Activity 5.6: Presentations and Critiquing
Time: 5 hours
Description
At various stages during the course, students are given opportunities to practise the skills of presenting
and critiquing. The formative assessment provided by the teacher and peers facilitate success when
culminating work is expected.
Strand(s) & Learning Expectations
Ontario Catholic School Graduate Expectations
CGE5e - respects the rights, responsibilities, and contributions of self and others;
CGE5g - achieves excellence, originality, and integrity in one’s own work and supports these qualities in
the work of others.
Overall Expectations
DMV.01 - carry out a culminating project on a topic or issue of significance that requires the integration
and application of the expectations of the course;
DMV.02 - present a project to an audience and critique the projects of others.
Specific Expectations
DM1.03 - complete a clear, well-organized, and fully justified report of the investigation and its findings;
DM2.01 - create a summary of a project to present within a restricted length of time, using
communications technology effectively;
DM2.02 - answer questions about a project, fully justifying mathematical reasoning;
DM 2.03 - critique the mathematical work of others in a constructive fashion.
Planning Notes
Time must be allowed for presentations and critiquing throughout the course. Decide which activities to
use with the class and the number of student presentations of specified length for each activity.
Unit 5 – Page 22
 Mathematics of Data Management – University Preparation
Teaching/Learning Strategies
Students create a summary of their written report for their presentation since it will not be possible to
present their entire project. Key components of their project that should be part of the presentation
include:
 Introduction;
 Data and Data Analysis;
 Conclusions;
 Audience Questions .
Visuals (e.g., data projection unit, communication technology) should be used to make the presentation
effective (i.e., flows well and keeps the audience’s attention)]. Students should be encouraged to practise
their presentation once they have it planned to determine if the timing is appropriate. Students practise
speaking clearly, looking at the audience, welcoming questions from the audience, and responding to the
questions in an effective manner. Students work on anticipating questions that might arise and preparing
responses.
Critiquing student presentations may be new for many students, especially in the math classroom. The
teacher encourages students to critique the presentations/projects and provide positive feedback wherever
possible. However, the critiquer’s job is to look at the claims of a project and to respectfully challenge
the claims with questions about validity, appropriateness of the model, etc. This skill is one of the
expectations and should be assessed. The teacher evaluates a student’s critique of another student’s
project. This skill should not be evaluated using peer assessment.
Although teachers assess the presentations for mathematical content, it may be reasonable to have
students use a simple checklist (see Appendix D) or rubric (see Appendix C) when critiquing.
Assessment & Evaluation of Student Achievement
 The teacher assesses the presentation for mathematical content as well as effectiveness of
communication. (See sample rubric in Appendix C.)
 Several students should critique each project. The teacher evaluates critiquing skills by comparing
the teacher’s assessment of the project with the critique done by the student.
Unit 5 – Page 23
 Mathematics of Data Management – University Preparation
Appendix A
Rubric for Assessing Proposal for the Culminating Project Stage 1: Posing the Question
Category
Level 1
(50-59%)
- preliminary data
is minimally
appropriate for
the problem posed
Level 2
(60-69%)
- preliminary data
is somewhat
appropriate for
the problem posed
Level 3
(70-79%)
- preliminary data
is generally
appropriate for
the problem posed
Level 4
(80-100%)
- preliminary data
is thoroughly
appropriate for
the problem posed
Application
Presents
preliminary data
that is appropriate
for the problem
posed
- poses the
- poses the
- problem posed is - problem posed is
Communication
Poses a significant problem with
problem with
generally clear
thoroughly clear
problem to be
limited clarity
some clarity
considered in the
culminating
project
Note: A student whose achievement is below Level 1 (0%) has not met the expectations for this
assignment or activity.
Unit 5 – Page 24
 Mathematics of Data Management – University Preparation
Appendix B
Rubric for Assessing Mathematical Content of the Written Report
Level 4
(80-100%)
- always or almost
Communication
Uses terminology
always uses
and notation that
correct
is correct
terminology and
notation
- demonstrates
- demonstrates
- demonstrates
- demonstrates
Knowledge and
mathematical
mathematical
mathematical
mathematical
Application
Mathematical
procedures that
procedures that
procedures that
procedures that
procedures shown are rarely correct
are frequently
are generally
are always or
are correct and
and appropriate
correct and
correct and
almost always
appropriate
appropriate
appropriate
correct and
appropriate
- sometimes uses
most often uses
- consistently uses
Application Uses - rarely uses
technology
technology
technology
technology
technology
appropriately
appropriately
appropriately
appropriately
appropriately
- presents the
- presents the
- most often
- always or almost
Inquiry Presents
mathematical
mathematical
presents the
always presents
the mathematical
content in a
content in a
mathematical
the mathematical
content in a
sequence that
somewhat logical content in a
content in a
logical sequence
shows limited
sequence
logical sequence
logical sequence,
logic
showing all steps
clearly
- identifies some
- identifies most
- identifies all or
Inquiry Identifies - rarely identifies
assumptions and
of the
assumptions and
almost all
assumptions and
limitations of the
assumptions and
limitations of the
assumptions and
limitations of the
data analysis
limitations of the
data analysis
limitations of the
data analysis
data analysis
data analysis
- makes
- makes
- makes
- makes
Inquiry Makes
conclusions that
conclusions that
conclusions that
conclusions that
conclusions that
are somewhat
are generally
are consistently
are justified by the are justified in a
limited way by the justified by the
justified by the
justified by the
mathematical
mathematical
mathematical
mathematical
mathematical
analysis
analysis
analysis
analysis
analysis
Note: A student whose achievement is below Level 1 (50%) has not met the expectations for this
assignment or activity.
Category
Unit 5 – Page 25
Level 1
(50-59%)
- rarely uses
correct
terminology and
notation
Level 2
(60-69%)
- sometimes uses
correct
terminology and
notation
Level 3
(70-79%)
- generally usually
uses correct
terminology and
notation
 Mathematics of Data Management – University Preparation
Appendix C
Rubric for Assessing Presentations
Level 1
(50-59%)
- rarely uses
correct
terminology
Level 2
(60-69%)
- sometimes uses
correct
terminology
Level 3
(70-79%)
- generally uses
correct
terminology
Organization of
presentation
- presentation is
rarely clear and
logical
- presentation is
somewhat clear
and logical
- presentation is
most often clear
and logical
Mathematical
content
- presentation
rarely reflects
mathematical
understanding
- presentation
usually reflects
mathematical
understanding
Presentation Skills
- presenter’s voice
is rarely clear
- presentation
sometimes
reflects
mathematical
understanding
- presenter’s voice
is somewhat clear
Visual Aids
- visual aids are
rarely used to
enhance the
presentation
- visual aids are
sometimes used to
enhance the
presentation
- visual aids are
usually used to
enhance the
presentation
Communication
- ideas are rarely
stated clearly and
rarely connected
Communication
Use of
mathematical
terminology
- presenter’s voice
is usually clear
Level 4
(80-100%)
- always or almost
always uses
correct
terminology
- presentation is
always or almost
always clear and
logical
- presentation
reflects thorough
mathematical
understanding
- presenter’s voice
is always or
almost always
clear
- visual aids are
consistently used
to thoroughly
enhance the
presentation
- ideas are clearly
stated and the
ideas thoroughly
connect
- ideas are
- ideas are clearly
somewhat clearly stated for the most
stated and
part and the ideas
somewhat
mostly connect
connected
Audience
- presenter is
- presenter is
- presenter is
- presenter is
Questions
rarely able to
somewhat able to generally able to
consistently able
respond to
respond to
respond to
to respond to
questions from the questions from the questions from the questions from the
audience
audience
audience
audience
Note: A student whose achievement is below Level 1 (50%) has not met the expectations for this
assignment or activity.
Unit 5 – Page 26
 Mathematics of Data Management – University Preparation
Appendix D
Critiquing a Presentation
Presentation made by:
Critiqued by:
Topic:
Rate the quality of the presentation under the categories below using the following scale:
1-poor
2-fair
3- good
4- very good 5-outstanding
Introduction
Statement of the problem
Organization
1
1
1
2
2
2
3
3
3
4
4
4
5
5
5
Visuals
1
2
3
4
5
Communication
1
2
3
4
5
Technology
1
2
3
4
5
Conclusion
1
2
3
4
5
Response to Audience questions
1
2
3
4
5
The most positive features of the presentation were:
Questions I asked:
Questions I still would like answered:
Unit 5 – Page 27
 Mathematics of Data Management – University Preparation
Appendix E
Turing Program
var n : int % a random number
var c : int := 1 % a counter
var x : array 1 .. 100 of int
randomize
for i : 1 .. 100
x (i) := 0 % initialize the array
end for
randint (n, 1, 100)
x (n) := 1
for i : 1 .. 20 % for 20 days
for j : 1 .. c
randint (n, 1, 100)
x (n) := 1
end for
c := 0
for k : 1 .. 100
c := c + x (k) % add up the number of 1's
end for
put "Day ", i, " Number ", c
end for
C++ Program
#include<stdio.h>
#include<stdlib.h>
int main(void) {
int i, j, k, n, c=1;
int x[100];
randomize();
for(i=0;i<100; i++)
{ x[i]=0; }
n=random(100);
x[n]=1;
for (i=1; i<21; i++)
{ for (j=1; j<c+1; j++)
{ n=random(100);
x[n]=1; }
c=0;
for (k=0; k<100; k++)
{ c=c+x[k]; }
printf("\nday - %d num - %d", i, c); }
return 0;
}
Unit 5 – Page 28
 Mathematics of Data Management – University Preparation