Introduction
This is part of a series of second, fifth, and seventh grade Mathematics Program Reviews. This review includes a summary of the structure of the program, evaluations of a selected set of content areas, and evaluations of program quality. Ratings in these areas were made on a scale from 1 (poor) to 5 (outstanding). The overall evaluation was made using the traditional system of letter grades. For details of the methods used in this evaluation see Methods for Seventh Grade Program Reviews.
Student Text Structure
The student text structure includes:The fourteen chapters are arranged by mathematical topic.
The book begins with a 2 page "Introduction to the graphing calculator".
Each lesson is moderately clear on its goals, even if they are not explicitly stated, although some lessons mix related learning goals. Within each lesson there is a clear exposition, dealing generally with mechanics and concepts, on what is to be done to achieve a particular mathematical goal. Various mathematical properties are mentioned explicitly. Even the "mechanical" lessons contain an example of a word problem that is to be translated to an expression or equation and then solved by the manipulations in the lesson. The in class problems contain both thought/discussion exercises and a series of guide practice activities covering the material of the lesson. The problem sets for each lesson contain both "mechanical" problems and word problems, as well as mixed review problems keyed to previous lessons. Lessons are sometimes introduced with short items that are relatively disconnected from the key concepts and skills of the lesson, for example, some are introduced with calculator exercises. Word problems in both examples and problem sets are often topical, being based on current movies and their stars or current sporting events or sports figures. In addition to lessons, each chapter ends with a "highlights" page including vocabulary and key concepts, a study guide, and various assessment activities.
The book is moderately illustrated, with 31 extraneous pictures in the 50 pages from page 101 to page 150.
The glossary is moderately complete and is keyed to specific page numbers.
Content Area Evaluations
Properties, Order of Operations [5.0]
This book does an excellent job of presenting properties, order of operations and algebraic expressions. Manipulations with negative numbers, including powers of negatives, are included. The formal system of "cups and counters" for modeling algebraic manipulations can be omitted with no loss, as can the lessons on how to punch numbers into one model of one brand of graphing calculator.
Exponents, squares, roots [5.0]
This book brings in sophisticated algebraic manipulation early on. Product and quotient rules for exponential expressions are introduced and drilled with monomials as well as with regular numbers. Negative exponents are clearly described and used in a variety of evaluation and simplification problems.
Fractions [5.0]
This is an excellent, complete presentation of this topic. The technology sections are unnecessary and should be cut down or eliminated.
Decimals [5.0]
All decimal manipulations are clearly and thoroughly presented. The process of converting repeating decimals to simplified fractions is taught, which is unusual for this level but builds both number sense and skills in solving equations.
Percents [4.5]
There is an excellent focus on proportions as a way of teaching percents, essentially going back to the definition of percent as "per 100", such that students recognize that the question "x is what percentage of y" means x/y = %/100 (in older terminology, is/of = %/100). Most kids do well with this presentation, especially when it comes to word problems, and it leads naturally to understanding why percent is the same as the decimal x 100. Both methods of calculating percent are demonstrated with sufficiently complex numbers. A section on estimating percent is included in this chapter. It is appropriate and enhances the student's number sense. The calculator option for conversion and equation solving should be eliminated. A statistics section (under the heading "integration") disrupts the flow and should be removed.
Proportions [5.0]
Proportions are well treated in this text, starting with a clear statement of the property of proportions (a/b = c/d implies ad=bc implies a/b = c/d). Both the computational exercises and word problems are of excellent quality with sufficiently "complex" numbers. There is conversion within unit systems but not between them.
Expressions and Equations - Simplifying and Solving [5.0]
This is an excellent treatment of multistep equalities and inequalities containing negative numbers, fractions, and decimals. Simplification of equations using the distributive property and manipulations of both sides of the equation. Inequalities are presented in the examples and the problem sets. The properties of equalities and inequalities are correctly and clearly stated. The only shortcomings are the sections on use of "cups and counter" manipulatives at the start of the equation solving chapters and the use of calculators for computations as simple as 4.5n = 14. Both good number sense and a knowledge of fractions make a calculator inappropriate.
Expressions and Equations - Writing [5.0]
Students are taught to translate English sentences into mathematical equations (or proportions which become linear equations after cross multiplying) or inequalities. They are asked to approach word problems by writing "let" statements (let a = the number of apples) and then equations to be solved by multi-step transformation. This is a very impressive unit.
Graphing [5.0]
This book goes above and beyond what most pre-algebra texts offer. It defines function, domain, range, x and y intercept, and slope. The students are shown how to graph linear functions using slope and y intercept. Graphing systems of equations and inequalities are also given thorough treatment.
Shapes, Objects, Angles, Similarity, Congruence [4.5]
Almost all essential topics are well covered, with theorems, when they occur being clearly stated not just "explored" and left to the student for final statement. The presentation on constructions is weak, being covered in a relatively cursory manner in a single, brief "lab."
It is important to note that this rather good presentation occurs too late in the book. Many classes will not reach the material on this topic unless the teachers carefully choose which earlier lessons and activities are appropriate and which can be dropped, and also keep their classes moving through the course without interruption.
Area, Volume, Perimeter, Distance [4.5]
The material of this topic is well covered. All formulas are presented with derivations whenever possible. For example, the trapezoid area formula is reached by adding the areas of the two triangles formed by a diagonal of the trapezoid. Shaded area problems are introduced in the geometric probability section. While these problems have value solely as area problems, the addition of probability is not overly distracting. Surface area is presented as both "nets" and 3-D diagrams.
One disappointing part of this section is the use of calculators to do a large fraction of the basic addition and multiplication portions of the example problems, even things like 10 x 15 or 10 x 12 x 2 (p 633).
The Pythagorean theorem is given, without proof, on page 677. The page before contains a group "modeling with manipulatives" exercise on graph paper in which squares are drawn on each side of a 1, 2, radical 5 right triangle. The area of the squares on the legs of the triangle are easy to determine and the students are asked "what is the relationship between the areas of the three squares?" This cannot be determined from this diagram alone and in fact requires knowing the Pythagorean theorem which is not yet introduced. This exercise is a waste of time at this position in the text. Instead, they could have done a slightly different exercise in which students "derive" the Pythagorean theorem for individual special cases that can be done on graph paper or a geoboard. From there, it is a simple step not just to inferring a possible general rule, but to actually proving the Pythagorean theorem.
It is important to note that this rather good presentation occurs too late in the book. Many classes will not reach the material on this topic unless the teachers carefully choose which earlier lessons and activities are appropriate and which can be dropped, and also keep their classes moving through the course without interruption.
Program Quality Evaluations
Mathematical Depth [4.9]
In terms of content level, this book is first rate. Nearly all of the topics we might hope to see in the year before algebra are presented and covered to a high level. A student mastering the key material of this book will be well prepared to succeed in any algebra class. Indeed, students who master the material in this book will be a significant way through meeting the Algebra standards of some states, especially those states in which first year algebra material officially or practically ends prior to in depth coverage of quadratics.
Quality of Presentation [2.8]
This presentation rating attempts to balance what this book could be on the good side with and what it could be on the bad side. If proper choices are made, this book offers a uniformly high level of possible student learning, which is reflected in the high scores for every sub-topic and the high overall score. If poor choices are made, this book could be a waste.
This book really gets kids into algebra when it focuses on paper and pencil manipulations. On the other hand, when the book focuses on playing with manipulatives, tiles, and counters, or the use of calculators instead of minds, the preparation is less than good.
Other problems have to do with the length and the possibility that they key content will be displaced by other items. There are 739 pages of instructional text. Some of the material at the end is beyond the scope of what needs to be covered, but multiple key topics appear late in the book. For example, area and volume are covered in chapter 12, beginning at page 608, and right triangles, their properties and necessary concepts to deal with them, do not appear until chapter 13, beginning at page 662. In the absence of careful planning and triage of less important topics and inefficient activities, it is unlikely that the average class will get to these key topics. Thus, each and every teacher will have to make appropriate decisions about what to keep and what to omit. Two teachers with identical books could give drastically different preparation to their students just by the choices of what to emphasize, how many days to dedicate to long term projects, exploration activities, collecting data, playing with manipulatives or learning to do by calculator what should be done by brain power. The wrong choices will completely eliminate the quality of this book. Strong leadership may be necessary to make sure that this book is used to teach the excellent course that could be built from it.
Quality of Student Work [3.0]
This book has high mathematical content and the work related to the content is sufficient depth and scope to allow students to succeed. Ill use of less relevant lessons could lower the overall quality of the student work.
Overall Program Evaluation
The relatively high rating for this book reflects a very high level of content and a mixture of instructional strategies that are at least satisfactory. In the context of qualified teachers who make wise choices, this book would be a first rate choice for a class of students hoping to move on to a real algebra class in the next year. The level of content, with appropriate choices as to what to include, is capable of leading to an excellent preparation for top level students aspiring to a top level algebra class. With a reasonable and experienced teacher, this book could also succeed with less well prepared but capable students aspiring to take algebra in the year after this text is used.
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