Mathematically Correct
Algebra 1 Reviews

Methodology

Topic Selection

The review was accomplished by selecting a sample of major topic areas for evaluation. The major topics cover a good portion of the material expected in an effective course in introductory algebra. The major topics were elaborated by the definition of key subtopics within each major topic area. These subtopics were selected to cover most of the major topic area. Various standards and textbooks not under consideration were consulted in the selection of major topics and subtopics.

The selected major topics and their subtopics are given below.

• Linear equations in one variable
  • Transformation by addition and subtraction
  • Transformation by multiplication and division
  • Several Transformations
  • Variables on both sides
  • Linear equation related word problems

• Linear inequalities in one variable
  • Axioms of order - transformations producing equivalent inequalities
  • Compound Inequalities
  • Absolute value in open sentences
  • Solving Problems involving inequalities

• Linear functions
  • Equations in two variables
  • Points, lines, and their graphs
  • Slope of a line
  • Slope intercept form - Parallel and perpendicular
  • Determining the equation of a line
  • Functions defined by equations
  • Functions defined by tables and/or graphs

• Factoring and applications
  • Monomial factors of polynomials
  • Multiplying polynomials
  • Multiplying/factoring: Difference of two squares
  • Multiplying/factoring: Squares of binomials
  • Factoring trinomials (all types)
  • Factoring by grouping
  • General factoring
  • Solving equations by factoring
  • Word Problems

• Systems of linear equations and inequalities
  • Graphing 2 equations in 2 unknowns
  • Solving 2 equations in 2 unknowns by substitution
  • Solving 2 equations in 2 unknowns by linear combination
  • Problem solving with systems of equations
  • Graphing systems of linear inequalities
  • Systems of open sentences and linear programming
  • Systems of 3 equations in 3 unknowns
  • Matrix solutions to systems of equations

• Laws of exponents
  • Laws of exponents for products
  • Laws of exponents for quotients
  • Zero and negative exponents
  • Fractional exponents

• Radicals and radical expressions
  • Introduction (square roots, restrictions, irrational numbers)
  • Product and quotient properties
  • Simplifying radical expressions
  • Solving radical expressions
  • The Pythagorean theorem
  • The distance formula
  • N^th roots

Subtopic Evaluations

Each subtopic was further elaborated by the identification of features and expectations for content coverage. Notes on what should be covered, including sample terms to be defined, were used as a guide to the evaluation of each subtopic. These notes included sample exercises defining low, moderate, and high difficulty/achievement levels. Each subtopic was reviewed according to the notes for that content area. In addition to specifics as described in the notes for each subtopic area, the review for each subtopic attended to the dimensions outlined for major topics below. Finally, the review process attended to dimensions for overall evaluation of the entire text. These included:

Describing the use of technology

Noting emphases on proof, derivation, and mathematical justification

Recording the sequence of presentation

To illustrate the nature and depth of subtopic evaluations, a description of the notes for the evaluations within the major topic of systems of equations is given in the Appendix. This description elaborates points that were considered in the evaluation of a major topic and the embedded subtopics, along with a description of the criteria used to generate sample exercises at three levels of difficulty.

Major Topic Evaluations

Each major topic was evaluated on seven dimensions based on information gathered in the review of the subtopics within the major topic. The review for each major topic assigned a rating from 1 (poor) to 5 (outstanding) on each of the following dimensions:

Overall evaluation
An overall impression of the breadth and depth of student learning supported for the major topic. This evaluation should consider notes for each subtopic and particular strengths and weaknesses noted in the review.

Quality of presentation
A rating of the integration of the text and other presentation features into a structure that builds student understanding. This should reflect quality of each subtopic and their integration into the major topic.

Definitions of terms and explanations of concepts and procedures
Specific attention to the details for terms, concepts, and procedures, making sure the definitions and explanations are sufficient and clear. Notes for each subtopic should be reviewed to form a composite rating for the major topic.

Quality and sufficiency of student work
The student work should be sufficient to promote consolidation of the mathematical lesson. Mere counts of problems is insufficient to determine this, however, as the actual student processes must be considered.

Range of depth and scope in student work
The exercises and other student work should contain an adequate representation building from the easiest to the most difficult cases, and should cover all the salient aspects of the mathematical topics involved.

Quality and sufficiency of examples
Examples should be sufficient in quantity to demonstrate the topics well. They should explain not only procedures but offer rationale and justification at each step.

Emphasis on analytic methods
The power of the analytic methods of algebra should be emphasized over non-analytic methods such as using tables, guessing, or estimating from graphs.

Summary Evaluations for an Entire Textbook

To arrive at summary ratings for an entire textbook, the means of ratings for each of the seven major topics were computed. Overall ratings for three additional areas were assigned at this time based on notes from the reviews. These ratings were not assigned to individual major topics because they are intermittent across topic areas or they involve the sequencing of major topics as well as subtopics. The three additional rating areas were:

Logic and usefulness of presentation sequence
Considers the development within subtopics, the arrangement of subtopics within major topics, and the arrangement of major topics within the text as a whole.

Emphasis on proof, derivation, and mathematical justification
Most positive ratings reflect the elaboration of proof and derivations within the exposition and the expectation of proof by the student. Also considered is the extent of mathematical justification in examples and explanations.

Appropriateness of technology
When implemented, technology should be used in a way that supports student learning without substituting for learning or interfering with it. Detailed instructions in the use of particular technologies are to be considered a distraction from the mathematical focus of the lesson.