BC Education - Mathematics 10 to 12 Appendices

Technology Education IRP Appendix D: Assessment and Evaluation - Samples
Sample 8: Principles of Mathematics 11

Topic: Graphing Techniques

Prescribed Learning Outcomes:

Problem Solving

It is expected that the student will:

demonstrate the ability to work individually and co-operatively to solve problems
develop specific skills in selecting and using an appropriate problem-solving strategy or combination of strategies
use appropriate technology to assist in problem solving

Patterns and Relations (Relations and Functions)

It is expected that students will:

connect algebraic and graphical transformations of quadratic functions, using completing the square as required.
describe, graph and analyze polynomial and rational functions, using technology

In addition to these outcomes, the teacher assessed students' attitudes and group skills.

Unit Focus

The teacher emphasized the relationships between algebraic and graphical representations of functions to help students gain insight into the interconnectedness of various branches of mathematics. Students were given opportunity to study patterns and develop the graphing skills they needed to visually represent the abstract relationships described by equations. Students were given opportunities to demonstrate higher-order thinking skills as they determined the complex equations represented by graphs and learned to use graphing calculators.

Planning the Unit

To plan the unit, the teacher:

identified the prescribed learning outcomes for this unit
examined the prerequisite knowledge and skills needed to achieve these outcomes interpret function notation
determined which prerequisites were already in place and which should be reviewed
planned a variety of activities to help students achieve the learning outcomes and to connect their learning to other desirable outcomes (e.g., group skills, attitudes)
identified criteria to use to evaluate students' learning
designed assessment that was an integral part of the instructional process

The Unit

Demonstrating Understanding of Basic Functions and Their Graphs

Together, students and teacher created graphs of all seven basic functions.
Games were used to reinforce students' understanding of the basic functions (square route, rational, quadratic, linear, polynomial) and their associated graphs. In this manner, students received drill and practice until the teacher judged that the associations were automatic, and that all students were familiar with the basic functions and could graph them quickly.

Changing the Basic Equations

In order to develop transformations to the basic equations that would result in vertical translations, the teacher worked with the class to sketch graphs of formulae such as:
comparing them with the basic graph of

The teachers used these comparisons to work with students to develop the rule.
A similar process was used to develop students' understanding of, first, horizontal translations, then reflections, and finally dilations (expansions and compressions).
To further develop students' understanding of transformations, the teacher asked "what if?" questions. (e.g., What if ?) The teacher asked increasingly complex questions to challenge students' understanding.
Once all the transformations were identified, the teacher introduced combinations of them to the discussion, further increasing the complexity.

Reinforcing and Demonstrating Understanding

Students worked in small groups to determine the equations and tables of values that would result from various graphs, showing translations, reflections and dilations. Students used a graphing calculator to check their results.
The teacher set up a team competition where groups of students developed equations for another group to graph. These graphs were then exchanged with a second groups, whose task was to determine which equations were used. As a variation, groups exchanged a graphing calculator that displayed a graph, and challenged other groups to replicate the graph or determine the equations used.

Defining the Criteria

Mathematical Thinking

To what extend do students:

use their knowledge of the basic functions and the rules for transformations to draw the graphs associated with complex forms of the basic functions. For example:
predict changes that occur in a graph given variations to the equations for the basic functions
demonstrate the higher-level thinking necessary to reverse the activity (develop a complex equation when given its graph)
use appropriate mathematical terminology and notation
demonstrate the ability to input functions into a graphing calculator
compare graphs by explaining similarities and differences with respect to their defining equations
explain properties of graphs based on the mathematical properties of the equation (e.g., cannot be negative, therefore graph has restrictions)
matching a relation or real data to an appropriate function or graph.
demonstrate a willingness to explore, experiment, and use alternative strategies to make predictions and test hypotheses
show a willingness to persevere in solving difficult problems
show flexibility in the use of available resources (e.g., calculators, sketches, graphs, help from other students or the teacher)

Group Skills

To what extent do students:

communicate ideas clearly and understandably
work with other students in whole-class and small-group discussions, to build on ideas and reinforce understanding
present logical arguments to support their conclusions
listen to the ideas of other students

Assessing and Evaluating Performance

Observing and Questioning

The teacher assessed students' understanding, attitudes, and group skills informally throughout the unit by:

observing students as they participated in class and small-group activities, taking note of behaviors that indicated whether they were achieving the criteria established for the unit
asking questions to assess students' understanding of the concepts being taught
moving through the class, observing students' work, encouraging their efforts, and listening to comments as they worked in small groups (the teacher knew students were involved in the unit when they made comments like: "Let's see what happens if..."; "Maybe we should try..."; "I wonder what would happen if...").

Students completed a summary sheet designed to help them pull together and organize their learning in this unit. Students used available resources (e.g., notes, textbooks, in peer and teacher consultations) to complete this assignment. The teacher collected and reviewed the summary sheets, had students make corrections, and advised them to keep the sheets for future reference.

The teacher used the rating scale on the next page to evaluate the summary sheets.

Summary Sheet

Criteria	Rating
draws the basic graph associated with each form of the basic equation	1 2 3 4
uses a knowledge of the basic functions and the rules for transformations to draw the graphs associated with complex forms of the basic functions	1 2 3 4
identifies and plots key points or lines on the graphs	1 2 3 4
identifies each type of transformation that occurs as a result of identified changes to the basic equations	1 2 3 4

Key:

4-Exceptional
3-Acceptable
2-Needs improvement
1-Little understanding shown

Peer Assessment

The team competitions described in this unit was used as an activity for peer assessment. Each group kept records of the results of their team challenges. The teacher reviewed these records at the end of the activity. As an indication of the depth of students' understanding, the teacher took note of the complexity of the questions that students posed to other groups.

Self-Evaluation

Students completed the following evaluation of their group skills.

Assessing Group Skills

Self-Evaluation of Group Work

Based on your participation in your group, rate yourself on each of the following. Circle the number that corresponds with how satisfied you are with your performance in each area. Put a check mark in front of areas you would like help with in the future.

Need Help?	Criteria	Rating
	I listened to the ideas of others in my group.	1 2 3 4
	I communicated my ideas clearly and understandably.	1 2 3 4
	I helped to build on or summarize the ideas of other students in my group.	1 2 3 4
	I supported my conclusions and ideas with logical arguments.	1 2 3 4
	I encouraged other group members.	1 2 3 4
	I attempted to resolve conflicts that arose in my group.	1 2 3 4
	I participated in all phases of the activity.	1 2 3 4

Key:

4-Very satisfied
3-Satisfied
2-Need some improvement
1-Not satisfied

The teacher used two sections, "Social" and "Ideas," of the reference set Evaluating Group Skills Across Curriculum to evaluate the way individual students contributed to the success of their groups. Minimally acceptable was defined as scale point 2. The teacher rated the performance of each student by using information from observations made while moving among the groups, and from the students' self-evaluations.

Tool and Material Use

Group Communications Skills

Rating	Social Interaction	Ideas Development
5 Shapes the way the group works. Develops and extends the group's work in ideas and development.	The student is able to initiate, develop, and sustain interactions so that the group is able to work together harmoniously. The student frequently encourages the efforts of other group members, often asking them questions. The student is comfortable - but not driven - to provide leadership when needed, attempts to resolve conflicts among other group members, is able to let go of personal ideas to further group progress, and approaches the task with obvious enjoyment, fen accompanied by humour.	The student participates in all phases of the activity, although contributions vary according to relevant information or experience. The student provides constructive feedback, offers predictions and hypotheses, and poses intriguing questions. The student is able to offer clarification, elaboration, or explanations as needed, and builds upon and in some cases synthesized the ideas others offer. The student may use comparisons, analogies, examples, or humour to illustrate or emphasize a point.
4 Social interactions comfortable and well developed. Flexible and well developed ideas.	The student is comfortable working in a group and contributes to the social dynamics. The student may take a leadership role in organizing how the group will interact. The student takes responsibility for group processes by facilitating and extending discussions, and by persevering beyond initial solutions. The student is responsive to other group members and their ideas. The student tends to interact effectively with the group but may not have much effect on how the other group members work with each other.	The student contributes ideas, experience, and information that the group is able to use. The student may help to develop ideas by providing details, examples, reasons, and explanations. The student often makes suggestions, asks questions, or adjusts personal thinking after listening to others. The student may also rephrase, paraphrase, or pose questions as a way of challenging or building on ideas from other group members. The student is able to make relevant connections to other situations or ideas.
3 Socially engaged. ideas are appropriate and related to the task.	The student takes part in group discussions and follows the basic rules for working with others: taking turns, listening while others are speaking, and sometimes offering recognition or support. The student may ask for or offer required information. The student is willing to accept group decisions and may share some responsibility for how the group works.	The student contributes dome suggestions and ideas to the group. The student responds and sometimes adds to suggestions that others make, participates in brainstorming activities, shows interest in the ideas of others, and adds information. The student may not defend personal ideas, and tends to give in quickly when someone disagrees.
2 Inconsistent social interaction. Ideas often disconnected.	The student may begin to show some awareness of the responsibilities of contributing to a group, may sometimes recognize and respond to the needs and ideas of others, and may show appreciation or support. At other times the student may have difficulty taking turns or accepting suggestions from other students. The student may remain uncommitted, focusing on personal needs rather than the group's task.	The student contributes ideas that address the task, but these may be unconnected to the ideas of others. The student may offer several suggestions but appears unable to elaborate, explain, or clarify ideas. The student often related the activity to personal experiences by telling stories. The student may think out loud, judge others' ideas quickly, or drift off-task easily.
1 Largely unaware of others' needs Limited contribution of ideas.	The student may not understand how personal behaviour affects others, and may be disruptive, aggressive, uninvolved, or easily frustrated.	The student may remain silent throughout the activity, contribute one idea repeatedly, or recount personal experiences unrelated to the group task. The student may not acknowledge or add to the contributions of other group members, but may respond to direct questions or other prompts from adults.

The students' performances with respect to the safe and proper use of tools and materials was evaluated using the following performance scale:

Outstanding
The student makes exceptionally thoughtful choices of materials, uses tools with proficiency, is especially conscientious about the safety of self and others, and assists in the organization and maintenance of a safe, orderly work environment.

Competent
The student chooses appropriate materials and tools, uses tools correctly, uses the proper safety equipment and procedures, and exhibits personal preparedness with respect to clothes, shoes, hair, jewelry, sleeves, and so forth.

Unacceptable
The student may be able to identify common tools but is unsure what tools to use for particular tasks and materials. Materials choices may be inappropriate. The student may not use proper safety measures or may need excessive supervision in order to do so. The student may be unaware of how personal actions affect the safety of others.

Revised: September 2001

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