Appendix D: Assessment and Evaluation - Samples
Sample 3: Applications of Mathematics 12

Topic: Vectors

Prescribed Learning Outcomes:

Problem Solving

It is expected that students will:

• solve problems that involve a specific content area
• solve problems that involve more than one content area
• solve problems that involve mathematics within other disciplines
• analyze problems and identify the significant elements
• demonstrate the ability to work individually and cooperatively to solve problems
• determine that their solutions are correct and reasonable
• clearly communicate a solution to a problem and the process used to solve it
• use appropriate technology to assist in problem solving

Shape and Space

It is expected that students will:

• use appropriate technology to describe vector and scalar quantities
• assign meaning to the multiplication of a vector by a scalar
• determine the magnitude and direction of a resultant vector, using triangle or parallelogram methods
• model and solve problems in 2-D and simple 3-D using vector diagrams and technology

Unit Focus

The overall goal of the unit was to familiarize students with the use of vector analysis to solve physical problems. Assessment required students demonstrate their ability to read and comprehend problems whose components contain vector and scalar quantities, that they could translate those quantities into vector representations and that they could resolve the key resultant components in such a way that the solution to the problem(s) was effectively communicated.

Planning the Unit

To develop the unit, the teacher:

• determined the overall goals of the unit
• identified the related IRP outcomes to be targeted in the unit
• identified the prerequisite knowledge and skills students needed to achieve the targeted outcomes included in this unit, and planned for review
• looked for ways to connect students' learning with other desirable outcomes, including those associated with students' attitudes
• looked for ways to encourage students' understanding of the ways in which practical problems are solved mathematically
• planned a variety of integrated instructional and assessment activities to help students achieve identified outcomes
• determined criteria with which to evaluate students' learning for marking and reporting purposes. Whenever appropriate, the teacher involved the students in setting evaluation criteria.
• developed with students a rubric that would be used to mark the unit project and its presentation. The development of the rubric's criteria as done as early as possible and with the purpose of promoting the students understanding of the work that they were expected to accomplish.

The Unit

The purpose of the unit is to provide students with opportunities to use and assign meaning to vector and scalar quantities relating to real-world situations; use vector analysis to determine resultant vectors in situations involving both 2 and 3 dimensions; and use resultant vectors as solutions to practical problems.

To set the scene and review use of triangle trigonometry and parallel line rules, the teacher asked students to consider the development of a flight or watercraft simulator program that used velocities as integral parts of the game parameters. Examples might include a game that would help pilots or sailors estimate the effect of wind and/or current on the path of a plane or ship. The teacher pointed out the need to use formulas that would eventually appear in the computer program. Students noted that while they would not actually create the game program in this class, computing science students might wish to develop their work further.

Peer Assessment - Students were encouraged to visually check the vector analysis of other's work with a view to confirming both the vector representations of the problem and the solutions. In as much as possible, students were asked to confirm magnitude and directional appropriateness using head to tail and parallelogram rationale. In simple terms, students were encouraged to check that their peer' s drawings correctly represented the situations they were resolving.

Assessment Strategies - The teacher assessed the student's development of labeling and describing vector and scalar quantities. The teacher viewed the student's drawings and gave feedback on the appropriateness of their representations.

Developing Vector Analysis Skills

• The teacher had students "walk" their way through vector directions in such a way that they gained an understanding of what the term's magnitude and direction meant in a physical location sense.
• As students considered the effects of wind and current on the velocity of a boat, the teacher encouraged students to express the components as vectors.
• The teacher carefully made the point that a vector can be moved in a drawing as long as its magnitude and direction did not change. Students were shown that moving vectors allowed them to use simple trigonometry to find resultant vectors
• Students brainstormed possible navigation scenarios that required simple vector analysis.
• Small groups of students were assigned the task of writing descriptions of problems that required vector analysis to solve and would be part of the development of the game introduced earlier in unit. The groups then provided solutions to their problems. Groups then checked each other's problem sets and solutions.
• The teacher then selected problems that exemplified well constructed vector analysis situations and had students show they understood the fundamentals of vectors and their analysis by completing the questions. In as much as possible, the teacher pointed out alternative ways of finding solutions (e.g., scale diagrams or triangle solvers) to vector problems and then discussed the strengths and weaknesses of various strategies.
• Students applied their knowledge of vectors to navigational problems to develop formulas that might appear in the programming of a flight simulator or instructional interactive program. The teacher modeled simple examples such as finding the location of an aircraft flying horizontally on a heading of 285° that is being pushed by a wind from 195° as measured clockwise from north. Parameters such as an indicated speed of 300 km/h, wind airspeed given as 90 km/h (constant) and an elapsed time of 1 hour and 15 minutes were used to show the students the affect of the wind on the planes final location.
• Simple 3-D constructs were established by having students resolve situations such as a aircraft flying at 80 m/s due north and climbing at an angle of 20° to the horizontal. Students were asked to consider the effect of a horizontal wind blowing from west to east at a speed of 30 m/s on the aircraft's track as expressed as a vector.

Teacher Review - As the students grew in their understanding of using vectors to solve navigational problems, the teacher pointed out other uses of vector analysis including forces as expressed in newton's specifically as applied to structural designs such as trusses. In as much as possible students were asked to apply their learning to new situations.

Peer, Small Group and Teacher Assessment - Students were asked at various developmental stages to report their concepts in a small group setting. Members of the group were asked to provide feedback to the reporter.

The teacher created an environment that encouraged students to participate fully in the development of sample simulations and scenarios for the flight simulator model. The demonstration of student's ability to meet the prescribed outcomes, in individual terms, was encouraged by reporting on each student's ability to construct vector diagrams and use appropriate strategies to resolve them. Students were also required to effectively communicate the meaning of the resultant vectors imbedded in the solutions to most if not all analyses that were required of them.

Using Performance Activities in Instruction

To help students understand various ways of resolving vector problems using technology such as triangle solvers, each student was asked to report on the pro and cons of using various strategies and tools.

Defining the Criteria

Mathematical Thinking

To what extent did students

• Demonstrate an understanding vector representations of quantities
• Use vector analysis to solve problems
• Use technology to solve problems
• Use appropriate terminology in to describe problems and solutions

Case Study for Applications of Math 12

The following questions are based on the predicted effects of wind on a small plane heading due north at 150 mph. The planes altitude is 8000 feet.

1. A. Draw a scale "head to tail" vector representation of the effect of a 60 mph wind from the west on the plane above. Indicate scale of drawing and show resultant vector.

B. Find the effective ground velocity of the plane as exactly as possible.

2. Assume the plane slows to 120 mph and descends to 5000 feet at the request of air traffic control. The plane then turns to head due east and maintains a steady speed.

A. Draw a side view vector diagram that would show the effect of the plane flying through a down draft where the air is dropping at 30 mph. Find the value of the resultant vector. How many degrees from level flight is the planes projected path?

3. Assume the plane is redirected to head due south. It is now flying at 5000 feet at 120 mph. The plane then enters turbulent air. The air mass the plane is travelling through is moving up at 30 mph and is moving west at 60 mph.

A. Find the resultant vector that represents the turbulent air.

B. Show the vector that shows the planes resultant velocity whilst passing through the turbulent air. Indicate magnitude.

Unit Test

Students were given a carefully constructed test that initially required them to show an understanding of vector representations, terminology and the solution of simple resultant forms. Students were then required to show how formulas could be developed to find resultant vectors for relatively simple 2-D and 3-D situations. In as much as possible the initial modeling of the situations was related to the work the students had done on the navigation simulations while the final modeling required students to analyze new situations such as those involving forces on bridge components, for example.

Self Assessment

Students' assessment of the simulator project was based on understandings of how to find and apply resultant vector solutions. Students commented on their ability the way in which they used technology to solve vector problems.

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Revised: September 2001

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