Algebra: Structure and Method - Book 1

Section I - Organization and Features

The student text for Algebra: Structure and Method - Book 1 contains 784 pages organized into 12 chapters, although an additional section is essentially a 13th chapter. The chapters are arranged and identified by math topics, not by context topics.

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

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

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

There are a small number of 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 book contains an excellent presentation of the solution of linear equations by transformation. Prerequisite skills are built through the first two chapters and this topic is covered completely in chapter 3. Explanations are clear for each subtopic, topics grow appropriately in depth and complexity, and there are a sufficient number of problems at each difficulty level to allow students to master the topic up to the highest level they are capable of. Word problem skills are well developed starting in Chapter 1 and most subtopics are accompanied by appropriate word problems. There are mixed review sections of about 10 problems each interspersed throughout the chapter.

Some calculator exercises are included as separate sections between subtopics. These are generally weak but are unobtrusive with regard to the key points having to do with the topic.

Proof and justification are relatively weak in most subsections, although there is a separate subsection specifically directed at algebraic proof and justification. It would be better if the groundwork for this topic had been set down continuously through the earlier sections.

There is excellent coverage of this topic. The material is linked to the chapter containing linear inequalities in two variables and is separated from solving linear equations in one variable by a number of chapters (linear equations are in chapter 3 and linear inequalities are in chapter 10).

Problems cover the full range of difficulty and sufficient numbers of problems are included at each level. The properties of order are presented a bit abruptly and largely by assertion, although the rules are clear. There is little proof or justification of steps. There is only one section of word problems, but it incorporates a wide and appropriate range of understanding and skills with good development of appropriate sub-skills.

This is a very good to excellent treatment of this topic. Some of the presentations are a bit abrupt with less than perfect logical development of the concepts. In general, the descriptions of what to do and how to do it are well presented and the problems cover the range of difficulties. There is some inappropriate use of calculators in terms of mindlessly using graphing calculators to "check" the results of analytical methods. The section on the slope of a line has some interesting problems relating simple analytic geometry to slope. Point slope form, per se, is missing, although a method for determining the equation of a line given a point and the slope is developed.

There is one mathematically incorrect section with an inappropriate use of calculators. This deals with the "line of best fit" to a set of plotted data. This separate "Application" section tells students to just "draw the line that seems to fit the points in the graph" and "You may wish to use a computer program to draw the graph." The hand drawn line is not a line of best fit and implying that what is being done is making a line of best fit is incorrect. The calculator can draw a line of best fit, but the use as suggested is completely mindless and without understanding.

This is a mixed coverage of this topic. The problems cover the range and scope of the topic and include some interesting and thought provoking questions. Some sections are broken into appropriate sub-sub-topics, factoring by grouping is well presented and a clear strategy for approaching a generalized factoring problem is given.

The sections on solving equations by factoring and on solving word problems involving factoring cover key points well, including why we set equal to zero and factor to get the answer and the need to eliminate physically impossible solutions. On the other hand, a number of topics are poorly and overly rapidly developed and would require a substantial amount of sophisticated teacher presentation.

Some irritating examples:

Finding a common monomial factor is essentially dumped on students with no strategy for doing this efficiently.

Many sections begin with an algebra tile-like explanation which does not really clarify what is going on. This is particularly true for the difference of two squares.

Multiplying binomials appropriately notes that this is an application of the distributive property and that it can be done as a vertical or horizontal multiplication. One thing that can be done with the vertical representation is to stress that this is a general application of the standard multidigit multiplication algorithm, stressing both what is going on and the concept of the distributive property in multidigit multiplication. Unfortunately, these calculations are presented in a way that is opposite of what is done with numbers. The higher order terms are multiplied before the integer terms and the alignment of the intermediate products is such that the integer and x terms are not aligned with the integer and x terms being multiplied. This confuses things rather than clarifying.

There is poor development of continued factoring of problems such as x⁴ - y⁴.

This major topic is characterized by lots of clear examples and lots of problems, with medium problems well represented in the mix and some hard problems. The most outstanding feature, however, is the number and quality of application problems which appear in three different sections in large supply.

Graphing does not get to difficult problems and omits the terms consistent and inconsistent.

Linear combinations are treated in two sections for good coverage. Substitution has good coverage as well. The section on linear inequalities is to be commended for including a good number of more difficult cases.

The section on linear programming is very brief and has an insufficient number of problems. The treatment of 3 equations in 3 unknowns is not given separately. These problems appear in each of 3 distinct solution sections, but are not explained. There is no treatment of the matrix solution.

Requirements for proof and justifications based on properties are lacking.

This major topic is addressed with clear explanations, lots of detailed examples, and lots of student exercises. Terms, concepts, and procedures are all clearly defined. Fractional exponents are even covered to the high difficulty level. In fact, the highest difficulty level and an emphasis on proof and derivation are both well supported in this text. Thus, the major topic is addressed in a way that would support student achievement at high levels.

This text presents the major topic in detail with plenty of problems at all levels.

The early subtopics start out with easier problems as they stick to numeric cases for the introduction of square roots and the properties. Then, variables are introduced and the harder problems begin. Proof of irrationality of root 2 is given, and proof for root 3 left to the student.

The Pythagorean theorem and distance formula follow. Proof is not given but is alluded to.

The section on simplification is outstanding and covers many approaches in separate subsections over 9 pages with many problems and many examples. This makes the section on solutions much easier to deal with even for hard problems.

This is followed by a short section on n^th roots.

The only additions this treatment might benefit from are a proof of the Pythagorean theorem, some more complex problems related to properties, and a little more detail on n^th roots.

Section III - Overall Evaluation

Mathematical Depth and Breadth

The coverage of math topics is excellent. Nearly all specifics receive abundant attention, with the exception of the more advanced sub-topics in systems of equations.

The depth of coverage is outstanding. The support and experience that students need to achieve well beyond moderate levels is provided, although the highest achievement levels need a bit more support in some areas.

Overall the quality of presentation is excellent in terms of the depth of student learning supported. Some weakness was found in the presentation on factoring.

Terms, concepts, and procedures are clear and stated explicitly in nearly all cases.

Examples are very effective and useful. In some cases, however, the presentation is too abrupt and complete student understanding may be at risk. Other examples also go well beyond the moderate difficulty level.

The emphasis on properties is generally good, but providing and requiring proof is intermittent.

The use of technology is not very frequent and is unlikely to be presented without underlying understanding.

The number of student exercises is outstanding across topic areas, including an extensive number of application problems in may subtopic areas.

The student work consistently extends into the moderate difficulty range, and the higher levels of difficulty are often provided. Thus, the exercises provide excellent experience to support moderate levels of achievement and very good support for high levels of achievement

The book provides an outstanding opportunity for student learning. Even achievement at the highest levels is usually supported.

The material on the laws of exponents is especially well done. On the other hand, the section on factoring has room for improvement, and proof is treated sporadically. Nonetheless, nearly all ratings of this program are outstanding. It thus does a good job of the topic of introductory algebra.