Course Profile   Mathematics of Data Management (MDM4U), Grade 12, University Preparation, Combined

 

Unit 5:  Managing the Culminating Project

 

Activity 5.1 | Activity 5.2 | Activity 5.3 | Activity 5.4 | Activity 5.5 | Activity 5.6

 

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.

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.

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.

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).

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:

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 well-organized 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).

·         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.

·         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.”

Appendix 5.2.1

Student Datasheet

Group

L1980

L1981

L1982

L1983

L1984

L1985

L1986

“Avg inc”

47703

47479

46563

45451

45608

46526

47106

“Med inc”

42250

41716

40416

38895

39216

39613

40062

0-5000

2.8

2.2

2.3

2.6

2.5

2.3

2.2

5000-9999

5.5

5.3

5.1

5.9

5.3

4.9

4.5

10000-14999

8

8.3

8.5

9.3

9.2

9.1

9.2

15000-19999

6.9

6.6

6.8

7

7.3

7.3

7.6

20000-24999

6.2

6.5

7.2

7.6

7.7

7.6

7.4

25000-29999

5.7

6.3

6.8

6.5

6.6

6.7

6.7

30000-34999

6

6.3

6.2

6.3

6.3

6.4

6.5

35000-39999

6

6.4

6.5

6.4

6.1

6.2

5.9

40000-44999

6.1

6.1

6.3

6.2

5.9

5.9

6.1

45000-49999

5.7

5.8

6

5.7

5.9

5.8

5.6

50000-54999

6.1

5.8

5.4

4.8

5.7

5.3

5.2

55000-59999

5.4

5.2

5.1

5.2

4.9

5

5

60000-64999

4.6

4.5

4.4

4.2

4.3

4.2

4.1

65000-69999

4.2

4.5

3.9

3.5

3.7

3.8

3.9

70000-74999

3.7

3.5

3.3

3.1

3.2

3.4

3.4

75000-79999

3.1

2.6

2.8

2.7

2.7

2.7

2.8

80000-84999

2.6

2.5

2.2

2.3

2.1

2.3

2.4

85000-89999

2

2.2

1.9

1.9

1.8

2

2

90000-99999

2.8

3

2.9

2.8

2.7

2.8

2.9

100000-124999

3.8

3.9

3.6

3.3

3.5

3.8

3.8

125000-149999

1.5

1.5

1.5

1.6

1.3

1.5

1.5

150000 and over

1.4

1.1

1.2

1.2

1.3

1.2

1.5

 

Appendix 5.2.1  (Continued)

 

L1987

L1988

L1989

L1990

L1991

L1992

L1993

L1994

L1995

L1996

L1997

L1998

47420

48429

49913

49116

47487

47603

46416

47254

47246

47476

48124

49797

39975

4730

42455

41294

39137

39766

38061

39255

38608

38613

38325

39398

2

1.8

1.5

1.7

2

2

2.1

1.7

1.9

2.4

2.4

2.3

5

4.8

4

4.2

4.9

4.7

4.7

5.2

4.8

5.2

5.2

5

9

8.9

8.5

8.8

8.9

8.9

9.4

8.8

8.9

9.4

9.3

9.1

7.1

7

7.2

7.2

7.6

7.7

8.1

8.6

8

7.8

7.8

7.3

7.7

7.3

7.3

7.6

7.5

7.4

8

7.4

8.1

7.4

7.4

7.3

6.9

6.8

6.2

6.6

6.9

7.1

6.8

6.5

6.7

7.1

7

7.2

6.2

6.4

6.3

6.2

6.9

6.7

7.1

6.4

6.9

6.4

6.7

6.7

6.3

6.1

6.3

6.1

6.5

5.8

6

6.3

6.3

5.9

6.2

5.8

5.8

5.6

5.9

5.6

5.6

6.1

5.9

5.9

6

5.8

5.2

5.5

5.8

5.7

5.9

5.7

5.5

5.3

5.1

5.1

5.2

4.8

4.9

4.9

5.3

5.2

5.3

5

4.6

5

4.7

5.2

4.8

4.9

4.8

4.7

4.7

4.5

4.9

4.8

4.8

4.6

4.3

4.6

4.4

4.6

4.2

4.5

4.1

4.2

4.4

4.4

3.9

4.1

4.4

4.3

3.9

4.2

4.2

3.6

3.8

4.1

3.7

3.8

3.6

3.7

3.2

3.6

3.6

3.7

3.6

3.7

3.3

3.2

3.2

3.3

3.5

3.2

3.1

3.1

3

3.1

3

3.3

2.6

2.9

3.1

2.8

2.7

2.8

2.6

2.5

2.7

2.7

2.6

2.8

2.3

2.4

2.5

2.6

2.2

2.2

2.3

2.2

2.3

2.4

2.3

2.1

2

2.1

2.1

2.1

1.9

2.1

2

1.9

2

1.9

1.8

2

3

3.2

3.3

3.4

3.1

3.3

3.1

3

2.9

2.7

3.3

3.2

4

4.4

4.7

4.5

4.1

4.3

3.9

4.3

4

4.4

4.1

4.9

1.6

1.6

1.9

1.7

1.7

1.6

1.5

1.9

1.6

1.7

1.9

2

1.6

1.8

1.8

1.9

1.6

1.6

1.4

1.5

1.7

1.7

2

2.3

 

Year

CP1

1980

52.4

1981

58.9

1982

65.3

1983

69.1

1984

72.1

1985

75

1986

78.1

1987

81.5

1988

84.8

1989

89

1990

93.3

1991

98.5

1992

100

1993

101.8

1994

102

1995

104.2

1996

105.9

1997

107.6

1998

108.6

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?”

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

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

1433

1556

1732

1758

1733

1579

1063

688

599

415

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 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?

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 the higher probability: getting at least one “6” in four rolls of a die () ) or getting at least one double-six in 24 throws of two dice () ) . De Mere 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 .. This dice problem has since been known as de Mere’s Problem. A 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.

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)

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 A: 11,10,9,3,2,1

Dice B: 8,8,8,7,1,0                                              Dice B: 9,8,8,7,1,0

Dice C: 6,6,6,6,5,5                                              Dice C: 7,7,6,6,5,5

Dice D: 12,12,4,4,4,4                                          Dice D: 12,11,5,4,4,3

Some three dice non-transitive sets are:

Dice A: 4,4,4,4,1,1                                             Dice A: 7,7,5,5,3,3

Dice B: 3,3,3,3,3,3                                              Dice B: 9,9,4,4,2,2

Dice C: 5,5,2,2,2,2                                              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?

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 Necessary

Combination 1

Comb. 1 Wins

Combination 2

Comb. 2 Wins

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?

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)

5 Games

Teacher

 

 

Student

 

 

No. games

5

 

No. games

5

 

P(wins game)

0.667

 

P(wins game)

0.333

 

Probability teacher wins in:

Probability student wins in:

5

0.132

 

5

0.004

 

6

0.219

 

6

0.014

 

7

0.219

 

7

0.027

 

8

0.171

 

8

0.043

 

9

0.114

 

9

0.057

 

Total

0.855

 

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.

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.

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.

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:

   

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;