Appendix
D: Assessment and Evaluation - Samples
Sample 9: Principles of Mathematics 12
Topic: Logarithms
Prescribed Learning
Outcomes:
Problem Solving
It is expected that the
student will:
- analyse a problem and
identify the significant elements
- develop specific skills
in selecting and using an appropriate problem-solving strategy or combination
of strategies chosen from, but not restricted to, the following:
- guess and check
- look for a pattern
- make a systematic list
- make and use a drawing
or model
- eliminate possibilities
- work backward
- simplify the original
problem
- develop alternative original;
approaches
- analyse key words
- demonstrate the ability
to work individually and co-operatively to solve problems
- determine that the solutions
are correct and reasonable, and clearly communicate the process used to solve
the problem
- use appropriate technology
to assist in problem solving
Patterns and Relations
(Relations and functions)
It is expected that students
will:
- model, graph, and apply
exponential functions to solve problems
- change functions from
exponential form to logarithmic form and vice versa
- model, graph, and apply
logarithmic functions to solve problems
- explain the relationship
between the laws of logarithms and the laws of exponents
In addition to these outcomes,
the teacher assessed students' attitudes, and group and communication skills.
Unit
focus
Real-life problems in such
divers situations as investment analysis, population growth, and radioactive
decay cannot easily be solved using basic algebra. This unit focuses on building
students' understanding of the inverse relationship between exponential and
logarithmic functions so they can find solutions to such problems. The teacher
provided opportunities for students to work individually and in small groups
to demonstrate their abilities to solve application problems and their knowledge
about why logarithms are important and how they can be used.
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
exponent laws and operations with exponents with the students
The Unit
Understanding Base
10 Logarithms
- The unit began with a
review of exponent laws and operations with exponents.
- The teacher worked with
the students to develop the definition of base 10 logs. Using numeric examples
to establish students' understanding, the teacher guided them to discover
the pattern:
The teacher related students'
discovery and understanding of the pattern to the formal definition of base
10 logs, and explained that log was really just another name for exponent.
- To move students toward
working with base 10 logs of numbers that are not integral powers of 10, the
teacher asked students to estimate, based on their knowledge of base 10 logs,
what power 10 would be raised to in order to get the log of a number such
as 65. The teachers asked questions to prompt their thinking (e.g., What is
it between? Is it closer to 1 or 2?).
- Students were asked to
find a button on their calculator that would help them find the log of 65.
Together, the students and teacher worked through a number of examples, first
estimating what the log might be, then using their calculators to find the
actual log.
- To develop the restriction
to the definition:
log x is defined only if x > 0 (the log of zero and negative numbers is
undefined)
students were challenged to guess the logs of one and zero and then find them
on their calculators. The teacher asked questions like:
- Why does zero give
you an error message on your calculators?
- Can 10 raised to
some power give you a negative number? Why not?
- Can you find the
log of a negative number?
- The teacher used similar
strategies to develop other rules associated with logarithms, such as:
Students were given practice
moving between exponential and logarithmic forms using their calculators before
proceeding to the next part of the unit.
Performing Operations
with Logarithms
- The class discussed why
it is important to be able to establish the same base for numbers. (Logs allow
us to establish a similar base for numbers and to work with them more easily
as a result.) The teacher asked specific questions that made use of exponent
properties. This reminded students about their experience working with exponential
expressions in algebra and of the exponent laws for multiplying and dividing
expressions with the same base. The teacher prompted students to make a connection
between their experiences with algebra and what they were learning about logs.
- The teacher used simple
concrete examples, such as:
as students began to experiment with the rules for multiplying logarithms
students worked with increasingly complex examples, such as:
(750)(83) = log a + log b
until they were ready
to develop the formula:
log(ab) = log a + log b
Demonstrating Change
of Base
- The class moved from
base 10 to other bases. The teacher used concrete numeric examples to help
students see the connections and relationships between exponents and logarithmic
forms. For example:
- By increasing the complexity
of the examples, the teacher helped students generalize to:
The teacher stressed that the base and the argument of the function (b & a)
must be greater than zero, and 
.
- Students confirmed their
understanding by working through examples that required them to move back
and forth between exponential and logarithmic forms.
- The teacher conducted
a brief discussion and provided examples to help students discover that the
rules for base 10 apply to other bases.
- The teacher challenged
students to find a log they could do on their calculators (e.g.,
and
then find a log they could not do on their calculators (e.g., 
).
Students learned to use the change of base formula
to solve the second type of problem. Students demonstrated their understanding
of these ideas by working individually on a number of prepared problems.
- The teacher worked with
the class to develop the graphs of exponential functions. Using graphing techniques
and the students' knowledge of inverses, the class developed a number of graphs
of exponential and logarithmic functions. Here are some examples:
Applying Knowledge
of Logarithms
To encourage students to
apply their knowledge of logarithms to real-life situations, the teacher asked
them to brainstorm subjects which interested them (e.g., money, earthquakes).
The teacher demonstrated how to use logs and exponents to solve problems related
to their interests. Students worked in small groups to solve a number of application
problems and then challenged each other with problems.
Defining
The Criteria
Mathematical Thinking
to what extend do students:
- demonstrate an understanding
of the definition of base 10 logs
- explain the for working
with logarithms
- describe why logarithms
are important and how they can be used
- describe the relationship
between exponents and logarithms
- apply the rules for multiplying
and dividing logs to solve simple and complex problems
- use the rules for logarithms
to solve problems involving exponential and logarithmic expressions
- find the logs of numbers
with bases other than 10
- use appropriate mathematical
terminology and notation
- use their calculators
to solve problems involving logarithms
- apply their knowledge
of logarithms to real-life problems
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., 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 extend 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
- calculators to solve
problems involving logarithms)
- 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 mathematical terminology
and notation and applied their knowledge of logarithms to real-life problems
Individual Projects
Each student completed a
research project on an issue or current event of personal interest. The project
required students to:
- use what they learned
about exponents and logarithms
- describe the topic and
why it was of interest
- describe how the use
of exponents and logarithms related to their topic
- look for data and display
their findings in a meaningful way
- 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 logarithms and exponents. 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 exponents and logarithms. 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
logarithms and exponents, 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
logarithms and exponents. The presentation is illogical and difficult
to follow. Findings are poorly organized and ineffectively presented.
References may be lacking or inappropriate for the topic.
|
Problem-Solving Observation
Sheet
As students worked individually
and in small groups to solve problems relating to logarithms, the teacher moved
through the class observing and asking questions to probe for understanding.
"the Problem-Solving Class Observation Sheet" included in the publication Evaluating
Problem Solving Across Curriculum was used to evaluate students' problem-solving
skills. The teacher used only those parts of the checklist that were appropriate
to the classroom activities. Students received a summary of the information
compiled on the sheet to give them feedback concerning their problem-solving
skills and techniques.
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 logarithms and use the rules to solve numeric 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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(at this point, students should recognize why having a common base is important).
The teacher and students worked together to see how to solve this problem.
They used these and similar examples to develop the process of taking logs
of both sides of an equation to generate solutions. Students demonstrated
their understanding by solving additional complex equations using their calculators.