Sample 1: Mathematics 8

Topic: Data Analysis

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 Concepts)

It is expected that students will:

• represent and apply fractional percents and percents greater than 100 in fraction or decimal form and vice versa

Statistics and Probability (Data Analysis)

It is expected that students will:

• select, defend, and use appropriate methods of collecting data:
  - designing and using surveys
  - research, using electronic media
• display data by hand or by computer in a variety of ways

In addition to these prescribed learning outcomes, assessment and evaluation also focussed on communication skills.

Unit Focus

The teacher's overall goals for the unit were to extend students' knowledge of concepts related to data analysis and to help them understand the usefulness and applicability of these concepts. Assessment and instruction were interwoven in the unit so that one led naturally to and informed the other. The teacher designed activities to assess students' problem-solving and communication skills and their understanding and application of concepts.

Planning The Unit

To develop the unit, the teacher:

• determined the overall goals for the unit
• identified the related IRP outcomes to be targeted in the unit
• examined the prerequisite knowledge and skills students needed to achieve these outcomes and planned for their review
• looked for ways to connect students' learning to other desirable outcomes, such as those associated with communication skills
• planned a variety of integrated instructional and assessment activities to help students achieve identified outcomes
• determined criteria with which to evaluate students' learning
• planned to evaluate students' learning for marking and reporting purposes and to inform future instruction
  

The Unit

Recalling, Reviewing, and Extending Relevant Concepts:

• The teacher checked students' knowledge of percents, decimals, and fractions, providing instruction and practice as needed to ensure that they were prepared to complete the activities included in the unit. The teacher also helped students recall what they knew about different types of charts and graphs.
  
  
• To relate students' prior knowledge to material included in the unit, the teacher gave pairs of students a set of data shown as decimals and instructed them to convert it to percents and then to fractions. Students chose two forms of the data and prepared a graph for each on poster paper, using any types of graphs they wished. Students hung their graphs around the room and compared them, discussing the effectiveness of the different types of graphs for displaying different types of data and the effects of differences in scaling. The class concluded that the proportions shown on the graphs were the same, regardless of the type of graph used or the form of the data set graphed. The teacher helped students discover how graphs could be misleading and how they could be used to misrepresent data.
  
  
• Formative Assessment-As students worked on this activity, the teacher moved around the classroom checking the accuracy of their conversions and graphs, providing assistance as needed. When the graphs were posted, the teacher challenged students to identify what errors had been made and how to correct them. During discussions, the teacher took note of the comprehension shown by students' comments and asked questions to stimulate and direct their thinking.
  
  
• The teacher showed students other types of charts and graphs that could be used for organizing and presenting data. The class discussed the appropriateness of each for use with different types of data.
  

Assigning Performance Activity Research Projects

• Students were asked to work in pairs to complete one of two research projects. The teacher gave each pair of students a detailed assignment sheet outlining the steps required for the completion of their projects.
  • topic 1: Prepare an itemized budget showing the average (mean) per-person costs associated with a 10-day family holiday to Disneyland.
  • topic 2: Prepare an itemized order sheet for a school store.
    
    
• As a class, students brainstormed budget categories for each topic and then categorized their ideas under major headings. For topic 1, these included food, drinks, games, rides, clothes, movies, souvenirs, and so on. For topic 2, the headings included pop, chips, bubble gum, school supplies, and so on.
  
  
• For topic 1, students were required to:
  • make an estimate of the personal amounts of spending money required for each category
  • make an estimate of the mean budget per item for the class members doing topic 1
  • collect budget items from at least three other class members
  • organize their personal and classmates' data into tables
    
    
• For topic 2, students were required to:
  • make an initial estimate of the costs of the initial stock for each type of item
  • research the net cost of items for the school store
  • survey a sample of the school population to determine the representative average (mode) number of items that the sample would consume in a school week (They also surveyed the sample to find the average-mean and median-the sample would be willing to spend in the school store each week.)
  • organize their data on tables showing all stock items as per-week amounts and the projected purchasing amounts
    
    
• For topics 1 and 2, students were required to:
  • use at least two different types of charts or graphs to display their data (Students chose the forms of the data-e.g., percents, fractions-they wished to display.)
  • write a brief summary that described data-collection methods, findings, and conclusions
    

Topic One: Holiday Project

Estimated	Personal	Mean	Fraction of Total Expenses
Drinks
Rides
Souvenirs
Clothes
Games
Food
(and so on)
Total

• The class talked about where they could find information required for the completion of their projects (e.g., travel agents, the Internet, banks, classifieds) and ways they might organize and display their data (e.g., stem-and-leaf plots, circle or bar graphs).
  
  
• Students presented the results of their research projects to the class for discussion and review. Presentations included a description of the methods used for data collection and a summary of findings and conclusions. The teacher worked with the class to identify errors in students' work.
  

Extensions - Using the Results of the Performance Activity in Further Instruction

The teacher worked with the class to create tables for both projects that compiled all of students' data for selected budget items. This information was then used to introduce measures of central tendency (mode, median, and mean). The class discussed the advantages and disadvantages of each measure when used with different types of data. The teacher used students' marks on the performance activity to reinforce these concepts. Students practised their skills by completing a variety of instructional activities that required them to determine the mode, median, and mean for different sets of data.

• The teacher introduced students to surveys, discussing their development and use. Students developed a survey to determine their schoolmates' perceptions of the cost of quitting school and living on their own. Students used the surveys to canvass the general student population and then compiled and discussed the results, comparing them to the class's findings regarding actual costs.

DEFINING THE CRITERIA

Mathematical Thinking

To what extent did students:

• recognize the relationship between percents, decimals, and fractions
• accurately perform calculations using rational numbers
• select, use, and defend effective methods of displaying, organizing, and graphing different types of data
• select, use, and defend appropriate methods of collecting data
• effectively present findings to the class and justify conclusions
• recognize sources of error

Communication Skills

To what extent did students:

• communicate ideas in a clear, logical, and understandable fashion to the teacher and other students
• justify and explain their reasoning and conclusions

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' understanding and communication skills informally throughout instruction by:

• observing students as they participated in class discussions and worked individually and in small groups, taking note of their flexibility in dealing with challenges, the effectiveness with which they used various resources to solve problems, and their ability to communicate and justify their ideas and reasoning in a clear and logical fashion
• asking questions to determine how well students understood the information they were learning
• looking for comprehension of the unit's 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:

• give students feedback regarding their progress
• determine the need for individual assistance
• regulate the pace of instruction
• determine the depth of information to include in the unit
• complete the marking scale for the performance activity

Evaluating the Performance ActivityResearch Projects

The teacher and students worked together to determine the criteria for judging students' research projects and presentations. The teacher made sure that the relevant defining criteria were included with the evaluation criteria for the project. The teacher organized the identified project evaluation criteria onto a rating scale, which was given to students before their presentations.

Evaluating the Research Projects and Presentations

Criteria	1	2	3	4	5
- used effective and appropriate methods of collecting data
- made accurate conversions among decimals, fraction, and percent
- made accurate calculation
- determined reasonable amounts for each category
- used effective and appropriate graphs for displaying data
- used appropriate scaling for graph
- clearly and adequately described data-collection mehtods, findings, and conclusions in the written summary
- clearly described the relationship between estimated and researched budget amounts in the written summary
- made valid conclusions that were clearly based on the data
- clearly and logically presented research and findings to the class
- used effective methods to organize and display data and findings for the presentation
- provided adequate justifications for methods, findings, and conclusions
- comminicated ideas in a clear, logical, and understandable fashion
- worked effectively with partner

Key:

5-Excellent
4-Good
3-Average
2-Needs Improvement
1-Needs complete review

Self-/Peer Evaluation

Students used the preceding rating scale as a guide for asking questions and providing feedback to their peers during their presentations. Following their presentations, each pair of students used the same rating scale to evaluate their own work on this project.

Teacher Evaluation

Students submitted their project summaries, data charts, graphs, and calculations to the teacher for marking, along with a description of the role played by each student in completing the project. The teacher used the rating scale to evaluate students' work. The teacher compiled the results of the ratings and calculated a class mean for each item. This information was used to identify areas needing additional instruction and to modify the unit for future use.

Improvement Plans

Based on feedback provided by their peers and the results of the self- and teacher evaluations, each pair of students completed an improvement plan. Improvement plans identified areas of strength and weakness and described what might have been done differently to improve the work on the project. Improvement plans were marked for:

• accuracy in identifying weaknesses (5 points)
• thoroughness (5 points)
• feasibility and effectiveness of proposed improvements (10 points)

Marks for the improvement plans were combined with those for the research projects to determine students' total marks for the unit.

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