Algebra 1: Explorations and Applications

Section I - Organization and Features

The student text for Algebra 1: Explorations and Applications contains 675 pages organized into 12 chapters. The chapters are arranged and identified by math topics, not by context topics.

The student text contains an index with a moderate number of entries. Index entries contain many context references, not just references to math topics.

The student text also contains a glossary with a small number of entries. The entries in the glossary include page number references. The breadth of coverage of mathematics terms in the glossary is moderate.

There are many answers to problems for students to check their own work.

There are many pictures within the text beyond those that clearly illustrate the material being presented.

The student text includes self-testing sections.

Section II - Major Topic Summaries

A) Linear equations in one variable

This is a fair treatment of this topic. The exposition is reasonable, terms are defined and concepts and procedures are explained moderately well. There is little use of calculators. On the other hand, there is no mention of either the addition/subtraction or multiplication/division properties of equality.

The problems in general reach only to a low or moderate level of difficulty. The variety of word problems and examples is quite limited. Few skills in equation writing are built from this material.

This is a completely unsatisfactory treatment of this topic. There are no compound inequalities and no absolute value inequalities, with the exception of a single "challenge" problem. The student work seldom exceeds the "easy" level. The properties of equalities and inequalities are not named in the text or in the list of "algebraic properties" at the back of the text. There are not sufficient examples. The word problems have only one problem of moderate difficulty. The student work has very little requirement for writing and solving inequalities.

This topic is treated poorly. The exposition appears to be the worst of all possible worlds. Subtopics begin with an exploration that is not much more than playing with a calculator. This is followed by a statement equivalent to "do it like this" with no logical development of why to do it that way or why it works. Most sub-sections lack any problems beyond the "easy" difficulty level.

Calculators are used mindlessly when real understanding would make the use of calculators unnecessary. For example, on page 284 there is a whole group of problems on converting equations from standard form to slope-intercept form, followed with an instruction to graph the equations, "Use a graphing calculator if you have one." This is absurd. A major advantage of slope-intercept form is that it leads to rapid graphing. Once in slope-intercept form, graphing should be immediate without technology.

This is a poor treatment of this topic. Problems seldom exceed the most basic level of difficulty. Factoring by grouping is missing. Students are given the quadratic formula and introduced to the discriminant two chapters before they learn to factor, precluding an understanding of how the quadratic formula is proven and leading to a habit of plug and chug for solution even when factoring would be easier and faster.

Cases in which use of the quadratic formula is encouraged seem to implicitly suggest the use of a calculator for problems quickly solved mentally. There is no obvious discussion in the exposition of factoring the difference of two squares or a perfect square trinomial, although there may be a small number of problems related to these methods. The word problems contain lots of spreadsheets, charts, graphs and creative writing, but few problems that emphasize writing and solving equations. Section 10.2, Multiplying Polynomials, is overwhelmed by a large exercise in paper folding, a pointless waste of time.

The presentation is of this major topic is seriously deficient. Graphic solutions are presented along with substitution in the first section. Perhaps, as a result, neither is given sufficient attention. Linear combinations are addressed in two sections, but even this much emphasis is not sufficient. References to properties of equality are not even given. No specific section on problem situations is provided and they are not handled sufficiently in the examples. Without this feature, the ability to apply the mathematics of this major topic is not well-supported. Difficult problems are extremely rare. Linear programming, matrix solutions, and 3 equations in 3 unknowns are not covered.

The number of problems for specific subtopics is often extremely limited. Problems are almost always of an easy variety. Explanations are not sufficiently detailed. There are some activities that are not efficient uses of time in terms of the learning potential they possess.

It is difficult to imagine students achieving competence at even moderate levels from this text on this major topic.

The text treats the properties of exponents in a simplistic way. Typically, simple numeric cases are given followed by a statement of a rule. The presentation is not sufficiently comprehensive. Student work is generally not sufficient and at very low levels of difficulty.

On the positive side, properties are stated as rules and given names so that at least students may refer to them. Also, a small number of problems do involve variables in expressions to be simplified.

Fractional exponents are not addressed.

The prospects for student learning of this major topic are poor.

The major topic is largely missing from this text. Some introduction to square roots appears, but the associated exercises are both too few and too easy.

This major topic begins with a statement of the Pythagorean theorem, requiring definitions of right triangles, perfect squares, square roots, the radical sign, and positive and negative square roots to be stated quickly in context without elaboration. One simple numeric example is given, followed by the statement of the converse of the Pythagorean theorem, and another simple numeric example. Then, a "Connection: Geometry" panel appears in which students illustrate the Pythagorean theorem by areas on graph paper for one particular case. Pythagorean triples are defined in one of the problems.

A short section on rational and irrational numbers and an example that shows how to convert a repeating decimal to a fraction in lowest terms then appear. The next section of the chapter states the product and quotient properties of square roots. These properties are not given names, so students will not be able to refer to them in their work and communication. Then the criteria for simplest form for radical expressions are stated. One example illustrates the quotient property with numbers, wherein the denominator is a perfect square. One example illustrates the product property with numbers. It happens to permit combining like terms since two instances of the square root of 6 appear. The process of estimating square roots to the nearest two integers is then given.

With this coverage, the simplification of radical expressions, even with numeric values, is not really addressed, since addition is only noted in passing. The solution of radical expressions is not addressed. Although the converse of the Pythagorean theorem is stated, the generalization to the distance formula is not addressed. Any generalization to radicands with variables is essentially missing.

The range of topic coverage might be considered appropriate for pre-algebra students, although the depth of coverage and the extent of student work might not even suffice for that application.

Section III - Overall Evaluation

Mathematical Depth and Breadth

The coverage of math topics is limited, meaning several specific details are not covered. The depth of coverage is generally inadequate, meaning that the support that students need to achieve beyond the most basic levels is not generally provided. The treatment of linear equations is actually good, but coverage for all other sampled major topics is poor.

Overall the quality of presentation is poor in terms of the depth of student learning supported. Terms, concepts, and procedures are sometimes addressed clearly, but other times the details are not addressed sufficiently or are missing entirely. Examples are not extensive enough or lack sufficient detail to support adequate depth of understanding.

The emphasis on properties, proof, and derivations is very low, making it difficult for students to comprehend the mathematical underpinnings of algebra. The use of technology is sometimes apt to interfere with student learning. The emphasis on analytical approaches is fair.

The number of student exercises is low, meaning that opportunities for consolidation of learning may not be sufficient. The student work does not cover a sufficient range of depth and scope. Student work is most typically at only basic levels. Thus, the presentation provides poor support for moderate and high levels of achievement.

The book provides a poor opportunity for student learning, with support for only basic levels of achievement. The appropriateness of the use of technology and the emphasis on analytic methods are fair, but the mathematics content coverage and depth are seriously insufficient.

The presentation style is poor in terms of the mathematical understanding supported, and the presentation and the practice exercises seriously lack sufficient breadth and depth.