Sample 2: Mathematics 9

Topic: Risk Management

Prescribed Learning Outcomes:

Problem Solving

It is expected that students will:

• use a variety of methods to solve real-life, practical, technical, and theoretical problems
• solve problems individually and co-operatively

Number (Number Operations)

It is expected that students will:

• solve problems, using rational numbers in meaningful contexts

Statistics and Probability (Chance and Uncertainty)

It is expected that students will:

• recognize that decisions based on probability may be a combination of theoretical calculations, experimental results, and subjective judgments
• demonstrate an understanding of the role of probability and statistics in society
• solve problems involving the probability of independent events
  

Unit Focus

The overall goals for the unit were to extend students' knowledge of concepts related to probability and chance and to help them recognize the need for developing effective methods for managing risk. Both assessment and instruction focussed on helping students understand where in their lives they might encounter probability and chance and on ensuring that they had the skills and under-standing needed to make good choices and informed decisions.

Planning The Unit

To develop the unit, the teacher:

• determined the overall goals of the unit
• identified the related IRP outcomes
• planned to review the prerequisite knowledge and skills students needed to achieve the targeted outcomes
• looked for ways to connect students' learning to other desirable outcomes, including those associated with group skills
• planned a variety of instructional and assessment activities to help students achieve identified outcomes
• planned to integrate instruction and assessment so that one informed the other
• established defining criteria to use to evaluate students' learning and determine their marks for the unit
  

The Unit

Recalling, Reviewing, and Extending Relevant Concepts

• The teacher used spinners and dice to review the Grade 8 concepts related to chance and probability. Students predicted the probable occurrence of specified single events. They then determined the number of times the events occurred during repeated trials and compared those numbers to their predictions.
  
  
• The teacher gave small groups of students decks of cards and asked them to predict the probability of drawing the seven of hearts. The students in each group then picked a card from the deck to see if anyone got a seven of hearts. This experiment was repeated several times, with students replacing the cards each time. The teacher challenged students to develop a method for determining the probability of choosing a seven on each successive trial. Students repeated the experiment additional times in their small groups, keeping track on a simple chart of the number of times a seven was drawn. The students discussed the results for each group and then combined their results for a class total. Class totals were compared to predicted probabilities to establish the relationship between theoretical calculations of probability and the actual frequency of occurrence of an event.
  
  
• The teacher used similar methods to help the class understand how to determine the probability of two independent events. The class practised their skills using cards, spinners, and pairs of dice.
  
  
• Formative Assessment-As students participated in these activities, the teacher moved through the classroom asking questions to check for understanding and providing additional instruction as needed. The teacher noted students' willingness to participate and to help each other as they worked in their small groups.
  
  
• Students also practised their skills by working individually, showing all their work, on a variety of problems compiled by the teacher from sources such as textbooks, worksheets, and the illustrated examples.
  
  
• Peer Assessment-Students exchanged their responses to the problems and checked each other's work using a key the teacher displayed on the overhead projector. Students were responsible for identifying the errors in each other's work. In addition, they had to describe how these errors could be fixed so the students who had done the problems could correct their own mistakes. The teacher collected and reviewed students' corrected work, looking for patterns of errors that might indicate a need for re-teaching.
  

Performance Activity 1 - Games of Chance

• The class discussed games of chance students may have encountered in their community, such as charity raffles and game booths at local fairs. They talked about how these games were used and what kinds of things had to happen for them to serve their purposes most effectively.

• Students worked in small groups to design their own games of chance. Each group was required to:
  • design a game that involved at least two independent events
  • provide a written explanation of the rules for playing
  • calculate the theoretical probability of winning their games
  • gather empirical data about their games through a minimum of 50 trials
  • compare their calculated probabilities of winning to the results of their trials to determine the fit
    
    
• Self-/Peer Assessment-Students tried out their games and directions for playing with other groups of students and refined them based on their own observations and feedback from their peers.
  
  
• The students turned the classroom into a midway and invited parents (and other students) in to play the games they had developed. Each player was given $50 in play money with which to play the games, and the class developed a payout scheme that was consistent for all games so that conclusions could be drawn from data resulting from the activity. A set fee of $2 was established as the cost of playing
  each game.
  
  
• During the activity, each group of students kept a running total of the number of times their game was played, the amount of money collected, and the amount paid out to winners.
  
  
• Teacher Review-The teacher moved among the games, evaluating their effectiveness, rating students' group skills, answering questions, and providing assistance as needed.
  

Using the Results of the Performance Activity in Further Instruction

• Following the midway activity, each group of students organized the data for their game and developed charts and graphs that would help them present their findings to the class. During the presentations, the class compared the success of various games and discussed the relationship between the predetermined probability of winning, the total amount of money paid out, and the accumulated profits.
  
  
• Formative Assessment-The teacher prompted students' thinking and measured the extent of their understanding with questions like:
  • What characteristics made some games more profitable than others?
  • What might have happened if participants could bet any amount they wished?
  • Based on this experience, what conclusions can be drawn about the games of chance you encounter at fairs and carnivals?

The teacher's formative assessment was based on their responses.

• The teacher helped students relate what they had learned from working with their own games of chance to major games of chance like a lottery. The class calculated and reviewed the probability of winning a major prize in a lottery.
  

Extending and Relating Learning

• The class discussed how risk and probability affect different decisions people make throughout their lives (e.g., where to go to college, what job to take, or how to invest their savings).
  
  
• An investment consultant from a local bank was invited to talk to the class about how probability and risk relate to various investment options (e.g., purchasing stocks, bonds, mutual funds, life insurance, real estate, or putting money into a savings account).
  
  
• Students worked in small groups on a brief, structured assignment to compare potential profits after 20 years on a $1,000 investment if they invested it in specific stocks, bonds, or mutual funds; put the money into savings; or bought $1,000 worth of lottery tickets (based on probability of winning). To help students with their calculations, the teacher provided tables they could use to determine interest and demonstrated the use of relevant computer programs. Students rated each investment option on a scale of 1 to 5 according to the amount of risk involved.
  
  
• Formative Assessment - Students discussed their responses to the assignment in class. The teacher reviewed students' work and listened to the discussion to determine if they were ready to complete Performance Activity 2, or if more preparation was needed.

Performance Activity 2 - Investment Project

• Students were told that they had each just inherited $10,000 and that the conditions for the inheritance required that they invest at least $5,000 of it for 10 years. Students were encouraged to gather information from a variety of sources (e.g., Internet, parents, school, community libraries, investment magazines, local banks, accountants) before making their investment decisions. Each student was required to:
  • account for what they would do with the entire $10,000
  • identify different investment options they might consider and specify the benefits and risks of each, considering the probability of return
  • develop an investment plan for the amount they decided to invest, specifying the investment choices they had made and justifying the reasons for those choices
  • speculate, based on what they had learned and on some basic research, how much money they would have after 10 years, specifying interest rates and showing all calculations where appropriate
  • specify where they had obtained their information
    

DEFINING THE CRITERIA

Mathematical Thinking

To what extent did students:

• determine the probability of two dependent events
• solve problems involving the probability of events
• describe the relationship between probability of winning, payout, and profit in games of chance
• describe how risk and probability relate to various investment decisions
• identify benefits and risks of various investment options
• choose and defend a strategy for investing a specified sum of money
• correctly perform calculations using rational numbers

Group Skills

To what extent did students:

• make positive contributions to the group
• build and elaborate on the ideas of others
• help develop the understanding of group members
• suggest positive ways to resolve differences of opinion among group members
  
  

ASSESSING AND EVALUATION STUDENT PERFORMANCE

The teacher designed assessment activities to ensure that decisions concerning students' learning were based on information from a variety of sources.

Observation and Questioning

The teacher assessed students' mathematical understanding and group skills informally throughout instruction by:

• observing them as they participated in class discussions and worked individually
• observing their work in small groups, taking note of their contributions to their groups, their success in building and elaborating on the ideas of others, their willingness to help other students, and their attempts to resolve any differences among group members
• asking questions to determine how well students understood the information included in the unit
• looking for evidence of understanding of and interest in central concepts in the questions students asked and in the comments they made to other students

The teacher used information obtained through observation and questioning to:

• provide feedback to students regarding their progress
• determine the need for individual assistance
• regulate the pace of instruction
• regulate the depth of information included in the unit
• help make decisions regarding students' marks
  

Evaluating the Performance Activity 1 - Games of Chance

The teacher used a marking guide to evaluate each group of students in relation to their performance in this activity. Students were given a copy of the marking guide before they developed their games.

The teacher evaluated members of each group individually using the following criteria:

• the student makes positive contributions to the group (3 points possible)
• the student builds and elaborates on the ideas of others (2 points possible)
• the student helps to develop the understanding of fellow group members (2 points possible)
• the student suggests positive ways to resolve differences of opinion among group members (2 points possible)
  

Evaluating Perfomance Activity 1 - Games of Chance

Criteria	Rating
the game involved at least two independent events	/3
the written explanation of the rules was clear, logical, and easy to follow	/5
students accurately calculated the theoretical probability of winning their game	/5
students performed at least 50 trials to gather empirical data about their game and recorded their observations	/3
students compared their calculated theoretical prbabilities to the results of their trials and accuately described the fit	/5
students effectively refiend their driections and game based on trials with other groups of students	/5
students effectively organized and displayed their game for use in the classroom midway	/5
during the midway activity, students kept careful records of the number of times their game was played, the amount participants paid for each play, and the amount paid to winners	/5
students accurately and effectively displayed their findings using charts and graphs and presented them to the class	/5

Evaluating Performance Activity 2 - Investment Project

• Peer Assessment-Students exchanged their investment projects with a partner and marked each other's work using a marking guide. Students were expected to explain the reasons for the marks they gave in each category and to specify what they felt their partners might have done differently to earn higher marks.
  
  
• Teacher Assessment-The teacher marked students' work using the same marking guide as used in the peer assessment. Students' final marks for the project were based on teacher ratings. Students' final marks for the unit were based on their marks for the two performance activities.

Student Survey

At the end of the unit, students were asked to complete a survey. The teacher used the results of the survey as a measure of the success of the unit and to modify activities for future use.

Evaluating Performance Activity 2 - Investment Project

1. What did you like best about the unit?
   ____________________________________________________________________________
   
   
2. What did you like least?
   ____________________________________________________________________________
   
   
3. What would you change?
   ____________________________________________________________________________
   
   
4. What was the most important thing you learned from this unit?
   ____________________________________________________________________________
   
   
5. Is there anything you would have liked more information about?
   ____________________________________________________________________________

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