Appendix
D: Assessment and Evaluation - Samples
Sample 10: Calculus 12
Topic: Limits
Prescribed Learning
Outcomes:
Problem Solving
It is expected that students
will:
- analyse a problem and
identify the significant elements
- demonstrate the ability
to work individually and co-operatively to solve problems
- determine that their
solutions are correct and reasonable
- clearly communicate a
solution to problem and justify the process used to solve it
- use appropriate technology
to assist in problem solving
Functions, Graphs
and Limits (Limits)
It is expected that students
will:
- demonstrate an understanding
of the concept of limit and notation used in expressing the limit of a function
f (x) as x approaches a:
- evaluate the limit of
a function:
- analytically
- graphically
- numerically
- distinguish between the
limit of a function as x approaches a and the value of the function
at x = a
- demonstrate an understanding
of the concept of one-sided limits and evaluate one-sided limits
- determine limits that
result in infinity (infinite limits)
- evaluate limits of functions
as x approaches infinity (limits at infinity)
- determine vertical and
horizontal asymptotes of a function, using limits
- determine whether a function
is continuous at x = a
In addition to these outcomes,
the teacher assessed outcomes related to the historical development of calculus,
students' attitudes, and group and communication skills.
Unit
focus
The concept of a limit plays
a central role in students future understanding of the importance and usefulness
calculus. It is this understanding that enables students to effectively interpret
the results of a variety of problem solutions using calculus.
Planning
the Unit
To plan the unit, the teacher:
- identified the prescribed
learning outcomes for instruction and assessment for the unit and the prerequisite
knowledge and skills needed to achieve these outcomes.
- determined which of these
prerequisites were already in place and which to review
- looked for ways to connect
students' learning to other desirable learning outcomes, including those associated
with group-work skills, communication skills, and attitudes
- identified the criteria
to use to evaluate students' success in the unit
- designed assessment to
be an integral part of the instructional process
- began the unit by reviewing
and discussing with students characteristics of functions that they were familiar
with (exponential, logarithmic, sinusoidal, etc.)
The Unit
Introducing the Concept
of Limit
- The unit began with a
review of the characteristics of familiar functions and their graphs.
- The teacher worked with
the students to develop the concept of limit using an overhead graphing utility
to demonstrate graphically the relationship between a variety of functions
and the limits of the function at a number of different points.
- Students used graphing
calculators to perform "what if" investigations to help them determine graphically
the effect on the limits of different types of functions if the characteristic
equation of the function were altered.
One-sided Limits
- Students were encouraged
to explore the concept of one-sided limits by comparing functions with one-sided
limits to those that had different left-hand and right-hand limits.
Infinite Limits and
Limits at Infinity
- The teacher provided
examples illustrating how to calculate limits of functions that result in
infinity and limits of functions as x approaches infinity.
- Guided practice reinforced
students' understanding of the difference between infinite limits and limits
at infinity.
Asymptotes of a Function
- The teacher then guided
class discussion from infinite limits and limits at infinity to vertical and
horizontal asymptotes of a function
- Students were given practice
determining the vertical and horizontal asymptotes of different functions,
using limits.
Continuous Functions
- The teacher began by
demonstrating that a continuous function is one which can be graphed over
each interval of its domain with one continuous motion of the pen.
- This concept was expanded
to the use of a graphing calculator where graphs of functions that were not
continuous at x = a had "holes" were no pixels were represented.
- Students were then asked
to use limits to describe this phenomenon. The teacher reinforced that a function
is continuous at a point
Applying Knowledge
of Limits
To encourage students to
apply their knowledge of limits to solve some of the historical problems that
the teacher had introduced at the beginning of the unit (i.e., tangent line
problem and the area problem). Students worked in small groups to develop possible
solutions to the problems.
Defining
the Criteria
Mathematical Thinking
to what extent do students:
- demonstrate an understanding
of the concept of limit
- can evaluate the limit
of a function in a variety of ways and show the connection between the methods
(analytical, graphical, numerical)
- describe why limits are
important and how they can be used
- describe the difference
between infinite limits and limits at infinity
- use limits to determine
asymptotes of a function and to determine whether the function is continuous
at a given point.
Attitudes
To what extent do students:
- approach problem situations
with confidence
- show a willingness to
persevere in solving difficult problems
- show flexibility in using
available resources (e.g., graphing calculators, textbooks, help from other
students or the teacher)
Group Skills
To what extent do students:
- work with other students
in whole-class discussions and small groups, to build on ideas and understanding
- initiate, develop, and
maintain interactions within the group
- help other students develop
understanding
Communication Skills
To what extent do students:
- communicate ideas clearly
and understandably
- listen to and make use
of the ideas of other students
Assessing
and Evaluating Student Performance
Observation and Questioning
Students' understanding,
attitudes, and group and communication skills were evaluated informally throughout
the unit. The pace of the unit was determined in part by the speed with which
students seemed to grasp the concepts.
The teacher:
- observed students as
they participated in whole-class and small-group activities looking for evidence
that they understood the concepts, built on ideas of others, initiated, developed,
and maintained interactions within the group, and assisted others in developing
understanding
- reviewed students' work
to note the extent to which they met the criteria for mathematical thinking
- made note of behaviours
that students were or were not achieving the criteria established for the
unit (e.g., using graphing calculators to determine limits when appropriate)
- used questions to assess
students' understanding of central concepts, the level of confidence they
displayed when problem solving, and their willingness to persevere when solving
difficult problems
- checked students' work
to note the extent to which they used appropriate limit terminology and notation.
Individual Projects
Each student completed a
research project on an topic or mathematician of personal interest. The project
required students to:
- use what they learned
about limits
- describe the topic and
why it was of interest
- describe how the use
of limits related to their topic
- explain what their finding
meant, why they were relevant, and why they were organized as they were
- use and cite a variety
of appropriate information sources
- complete a written report,
and present their findings to the class
Presentations and reports
were evaluated separately using the following holistic scale. Students received
copies of the scale before starting the project and used it to rate their presentations
and written reports. The teacher held conferences to discuss discrepancies between
students' and teacher's ratings. Students were given suggestions for improvements.
Students who received scores of one or two on their written reports were given
the opportunity to redo them. The final score for the written report was the
higher of the two scores.
Project
Rating Scale
|
4
- Outstanding
|
Information is presented
clearly, logically, and understandably. Examples or demonstrations are
used appropriately to illustrate explanations. Findings are well organized
and effectively displayed. Explanations indicate a clear understanding
of the topic and the use of limits in a variety of contexts (e.g., to
determine asymptotes of a function). References are appropriate for
the topic and indicate that the student understands where to look for
information.
|
|
3
- Adequate
|
The presentation
indicates that the student has a basic understanding of the topic and
of limits. Information is understandable. Findings are organized and
displayed acceptably. References are appropriate for the Topic.
|
|
2
- Needs Improvement
|
The presentation
indicates a limited understanding of either the topic or the use of
limits, or both. The presentation may be illogical or difficult to follow.
Findings may be organized poorly or ineffectively. References may indicate
that the student is not clear on finding the best sources of information.
|
|
1
- Inadequate
|
The presentation
indicates a lack of understanding of either the topic or the use of
limits. The presentation is illogical and difficult to follow. Findings
are poorly organized and ineffectively presented. References may be
lacking or inappropriate for the topic.
|
Test/Team Competition
To evaluate students' understanding
of the concepts developed in this unit, the teacher designed a team competition
based on completing a set of items on a study sheet. Students worked in groups
of four and used all available resources to answer each of the items. The groups
were given opportunities to drill each other. When all members of the group
felt comfortable with the information, they were asked to take a test individually
that contained items similar to those on the study sheet. The teacher collected
the tests, scored them, and gave each student a grade. Groups worked together
to correct their tests and returned their corrections for additional points.
The winning group had the highest average individual test scores and the highest
average scores on their corrections.
Take-Home Test
Students were given a take-home
test that contained similar problems to those studied during the unit. As part
of the test, students were required to explain the rules they had learned for
working with limits and use the rules to solve examples that illustrated their
explanations.
The last page of the test
included a self-evaluation sheet that asked the following questions:
- What things in this unit
did your find easy?
- What things did you find
difficult?
- Are there any areas in
the unit that you need more help with? If so, what are they?
- Would you be interested
in after-school peer tutoring for help in these areas?
- Would you be willing
to tutor another student after school in the areas you feel most comfortable
with?
The teacher used the results
of the take-home test and students' responses to the self-evaluation to pair
students for after-school peer tutoring. Students were required to correct errors
on their take-home tests and resubmit them for a second evaluations.
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Revised: September 2001
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