General Considerations
The ratings for this major topic should be based upon a review of the following subtopics:
In addition, the following subtopics should also be reviewed, but are not to be given primary consideration in the overall evaluation of the major topic:
Notes should be made concerning all major rating categories during the review of each subtopic.
Subtopic: Graphing 2 equations in 2 unknowns
This section should contain clearly worked out examples that illustrate the process both from equations and from context material.
It should be noted that the solution is the intersection point (if there is one), and that this point is a solution of the system since it satisfies both equations.
The fact that the graphic method is inexact should be clearly noted.
Good examples would show graphing from equations in different forms, or, better yet, manipulating the equations to facilitate graphing.
"Systems of linear equations", "a solution of the system", and "the point of intersection" should be defined. Including the term "simultaneous equations" is useful.
In this section, also evaluate the material that deals with the form of the solution, even if it is located in another area of the text. This material should clarify the possible outcomes, including no solution for parallel lines or infinitely many for coincident lines.
This should address terms including: "parallel", "coincident", "intersecting", "consistent" and "inconsistent", "dependent" and "independent".
Difficulty level for examples varies as a function of the form of the equations relative to graphing, the complexity of the equations or the contextual example leading to the equations, especially when fractions are included, and certain area problems.
Subtopic: Solving 2 equations in 2 unknowns by substitution
This section should show clear examples that note how the process is completed in increasingly complex cases.
The term "substitution" should be defined.
The material should emphasize that an expression is substituted for a variable.
Cases with coefficients on both variables should also be illustrated.
The relationship of this solution to the graphic method should be shown, and the accuracy of the analytic method noted.
Problem difficulty is a function of the complexity of the equations involved, and should include fractions. The highest level should involve other variables, so that the equations are solved in terms of these other variables. Problems that ask for proof are valuable in this section. This section lends itself to a wide variety of application problems.
Subtopic: Solving 2 equations in 2 unknowns by linear combination
This section may sometimes be called the method of elimination.
The presentation should begin with simple cases in which the addition or subtraction of equations will eliminate one term, and then follow with more complex situations in which one and then both equations must be multiplied by a constant first. The two methods may appear in separate sections that must be combined in rating this subtopic.
The simple case should refer to the additive property of equality, and the more complex should add a reference to the multiplicative property of equality.
The material should indicate that systems of equations before and after their transformation by the multiplicative property are equivalent systems.
Terms should include "equivalent systems", "linear combination", and "elimination".
Even the low level of difficulty should include cases requiring both methods of elimination. Moderate difficulty should involve fairly complex equations including fractions. Especially valuable are cases with variables in the denominator, and the method for solving these systems should be clarified. The highest level should include finding solutions in terms of other variables included in the equations. Proof is desired.
Subtopic: Problem solving with systems of equations
This subtopic will sometimes appear as a separate section. In other cases, it may be mixed into other subtopic areas and have to be evaluated by considering each of these. In exposition and examples, the presentation should address:
1) Identifying the question - Too often this is not reinforced. It is often important to identify exactly what is being asked for.
2) Assigning the variables - The habit of explicit statements of what the variables stand for should be encouraged, and this should be done in more complex cases.
3) Setting up the equations - Explicit statements of the equations should be given. In more complex cases, the meanings of the equations should be expressed.
4) Setting up equations in complex cases - Problem situations in which the equations are much less obvious from the situation should be included, especially those that involve unit changes and other manipulations to arrive at the equations.
5) Solving by an analytic method - Problem situations that lend themselves to solution by substitution and by linear combination should be included. Circumstances leading to a preference for one or the other method may be given here, although they may be covered in another section as well. Solutions should be worked in detailed steps, not just left to the student.
6) Checking the solution - Checks should regularly be employed.
7) Stating the answer - The answer should be stated explicitly, not left to be understood as just a numeric value (just including units is not sufficient).
8) Graphing and interpreting - In at least some cases, the solution should be illustrated graphically. The meaning of the solution should be interpreted in at least some cases.
Problem difficulty is largely based on the complexity of the equations themselves and their representation in the problem situation. Moderate difficulty should require some manipulation of equations prior to solution, and fairly complex problem situations. Coverage of many different contextual situations is needed. Hard problems will involve the most complex situations and equations, and should include problems that require the properties found in the classic "wind speed" and "current" problem varieties.
Subtopic: Graphing systems of linear inequalities
This section may be displaced from the other material on systems depending on the organization of the text.
Good exposition should include material that refers to important points in the general case of graphing inequalities, including the transformation of equations to forms convenient for graphing, notes on the use of dashed and solid lines, shading above or below depending on the relation.
The solution should be defined as the intersection of the shaded region. Special attention should be called to the fact that points on a (dashed) line may not be included in the solution.
The fact that all points in the shaded area are solutions should be indicated. It is useful to point to solutions for discrete cases where only whole numbers are viable. Cases that are restricted to the first quadrant should be noted with explanation. Solutions that extend beyond the grid of the graph should be noted. The interpretation of the solution should be illustrated with contextual applications.
Level of difficulty of exercises is a function of the number of inequalities involved and the complexity of these inequalities in terms of the number and form of their coefficients and the requirement for manipulation to facilitate graphing.
Subtopic: Systems of open sentences and linear programming
This subtopic will appear in various forms. The simplest form includes one equation and one inequality. Here the nature of the solution should be discussed and examples should be interpreted. More complex forms will have multiple inequalities and one equation to be maximized or minimized, and should be called linear programming.
Terms should include "corner points" or "vertices", "convex polygonal region", "feasibility" or "feasible region", and "constraints".
Low difficulty level problems will involve only one equation and one inequality. Moderate difficulty cases will involve linear programming with multiple inequalities involved. These must include cases with at least two inequalities that are not just horizontal or vertical lines. The highest difficulty level will include problem situations embedding the inequalities in a context.
Subtopic: Systems of 3 equations in 3 unknowns
This subtopic should apply the methods of substitution and linear combination to 3 equations in 3 unknowns.
Ordered triples and first degree (or linear) equations in three variables should be defined.
Examples should illustrate the process for both the substitution and linear combination methods.
Low difficulty problems involve simple equations in the same form and simple problem situations. Moderate levels will involve more complex coefficients and equations that are in different forms. The highest difficulty levels should involve other variables so that the solution is expressed in terms of these other variables, and may also involve finding systems yielding given solution sets subject to other conditions.
Subtopic: Matrix solutions to systems of equations
This subtopic may take a variety of forms.
One method involves laying out the coefficients in an array and multiplying rows and combining rows until coefficients are zero for some terms. This is a direct analog of the linear combinations technique.
Another method involves the use of the determinant in the solution.
The most complex method involves matrix inversion and is directly applicable to problems with more than two unknowns.
Whatever approach is used, the simple terms of linear algebra may be needed, such as matrix, element, coefficient matrix, augmented matrix, row equivalent, determinant, inverse of a matrix.
This topic should not introduce a cookbook approach to solutions without explaining the reasons why such a process works and showing the relationship to solutions by substitution or linear combination. Complex cookbook approaches, especially those using technology for matrix inversion, are at risk of being worse than not including this subtopic at all.
Simple problems involve questions about the definition and process involved. Moderate difficulty level involves the solution of simple problems by the matrix method selected. The highest difficulty level will involve solutions with equations that must be manipulated prior to the matrix solution and are not in the same form.