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 includes:The fourteen chapters are arranged by mathematical topic.
Each chapter is composed of about 8-9 lessons. Lessons are clearly laid out, and generally focus on clearly defined goals, with direct teaching of the skills or concepts to be learned. There are a 1-4 reasonable examples with good exposition, followed by a guided practice section reviewing the concepts and manipulations of the lesson. The exercises contain straightforward practice of the lesson plus word problems, as well as a few standardized test practice questions and an extension problem that is part of a "project" that continues through the chapter. The exercises can be quite challenging. About every second to fourth lesson, there is a "spiral review" incorporating problems from earlier in the book. Each chapter also comes with at least one exploration activity (some of which, such as modeling equation solving with algebra tiles, are not up to the quality of the rest of the book), a technology utilization lesson (most of which describe either how to punch numbers into a particular calculator or how to read the display), a mid-chapter assessment, a summary, a review and a final assessment. The summary, review, and assessments are clear and free of nonsense.
The book is moderately illustrated with 21 pages with extraneous pictures between pages 101 and 150. The pictures are generally small and not particularly distracting.
Content Area Evaluations
Properties, Order of Operations [4.0]
Except for too much reliance on calculators, this is an above average treatment of this topic. The commutative and associative properties are used in mid chapter 2 simplification problems. They are restated in a box at the foot of the page. Clearly, it is expected that this is review from the previous (grade 7) course. The distributive property is "introduced" at the start of chapter 2 with numerical values and variable expressions. Order of operations rules are drilled with positive numbers in chapter 1 and then again with negatives in chapter 3. This is a clear and sufficiently rigorous approach.
Exponents, squares, roots [4.0]
This is an excellent treatment of positive and negative powers, exponent roots, scientific notation, including both multiplication and estimating. There is coverage of factoring monomials including variables raised to powers and evaluating monomials with positive or negative exponents. The square root section is insufficient as it relies on calculators to provide the answers in far too many problems.
Fractions [5.0]
This is an excellent treatment of this topic. All fraction skills in this book are extended to monomials and monomial expressions, and both positive and negative fractions are presented. The amount of algebra that is brought in is impressive. The use of calculators is moderate and not too intrusive.
Decimals [5.0]
This is an excellent treatment of this topic for a WELL PREPARED student. Many of the skills of decimal arithmetic are assumed (multiply, divided, powers of ten, negatives and positives) and not directly taught. Surprisingly, these topics are also absent from the "Toolbox," although similar topics for fractions are covered. While it is refreshing to find a book that actually assumes that students have learned something prior to entering a class, if this book is used with a less than completely prepared class, it may be important to plan lessons on the basic manipulations of decimals.
Given the high expectations for students, the book can get right down to business with decimals, including not only converting terminating decimals to fractions and percents, but also converting repeating decimals to fractions as well. Indeed, this skill is well described and drilled. This is an excellent way to drill an aspect of number sense and algebraic thinking, but it is seldom taught.
Percents [5.0]
As with decimals, this is an excellent unit for students who come to the course fluent in the basics and are ready to solve a variety of problems using percents. This is not the book for remediating students up to a level ready for algebra, it assumes something was learned before. A complete chart of fraction/decimal/percent equivalents is presented early on. Percents are presented in the context of proportions ( x is what percent of y converts to x/y = %/100). This "is over of" form of presentation is clear to many students and invokes the definition of percent. It also leads directly to the "decimal times 100" form for calculating percents. In the case of this book, the presentation involves not the simplest of numbers, but rather "messy" numbers which challenge a student who is expected to be prepared. A number of word problems involve analyzing graphs of various types of graphs. These are appropriately placed. Other problems include sales tax, income tax, and population distribution. These are good "real world" problems with amazingly little technological enhancement.
Proportions [5.0]
Ratios, rates, proportions, similarity and scale drawings are all covered thoroughly and at a "pre-algebra" level. Customary and metric conversions and not presented as "topics" but they are thoroughly discussed individually when they come up in various geometry problems. There does not appear to be conversion between unit systems.
Expressions and Equations - Simplifying and Solving [4.5]
This is an excellent presentation brought down, slightly, by misguided use of modeling with algebra tiles. It is not clear that anyone need heavy usage of algebra tiles to learn to solve equations, but students who are ready for this coverage definitely do not.
The method for solving multi-step equations is clearly stated, with step-by-step examples. Fractions, decimals, negatives, grouping "like" terms, simplifying with the distributive property and variables on both sides are all presented in this excellent section. The number of algebra-type word problems is a bit thin and the problems invoking the skills mentioned above are mixed in with tables and technology exercises. Inequalities are presented in chapter 9, 5 chapters after equalities. The properties of inequality are clearly stated with a strong emphasis on the multiplication and division properties with negatives.
Expressions and Equations - Writing [4.0]
This is an above average treatment of this topic. "Translation" exercises and writing equations for word problems are presented at each stage of both the equation and inequality units. More practice is needed in writing equations for word problems. Not enough examples and practice problems are given. The "Problem Solving Strategies" section strays from algebra to charts and other non-analytical methods. The focus at this level should be on writing and solving equations and inequalities, not on other less efficient methods that one can use even without knowing anything about algebraic methods.
Graphing [5.0]
This book devotes a whole chapter to linear functions. X and y intercepts, slope, slope intercept form, and parallel lines are all defined. Students graph from tables, then from properties of the equation (e.g. slope, intercept). In one exercise very late in the book, they write equations in y = mx + b form, derive two points from this (one being the y intercept and the other being found from the slope) and plot. Students are taught to graph inequalities using a solid or dotted boundary line, and then testing points to determine where to shade as the solution area.
Data analysis finds its way into the word problems sets, but other types of problems involving straightforward linear relationships are also presented. In these, the student is asked to make an algebraic model (linear equation in ax + by = c form) and graph by plotting points. Since this is not an algebra course, they are not asked to go from this form of the equation to slope intercept form and then graph, but that is OK.
Shapes, Objects, Angles, Similarity, Congruence [4.5]
Other than the complete lack of constructions, this topic is covered in a well above average manner. All essential topics, and much more, except constructions, are presented. Points, lines, planes, classification of angles of polygons, classification of quadrilaterals including "kites",. and the angle/side inequality relationships for triangles (order of side lengths = order of measures of opposite angles) are presented and drilled in the exercises. Algebraic exercises are included using variables for angle measures. Formulas are derived and listed for the sum interior angle measures of polygons, for the measure of each interior angle of a regular polygon, and for the measure of each external angle of a regular polygon.
Some triangle and circle vocabulary seems to be missing (median, angle bisector, arc) but the overall content level is way beyond what would typically be called pre-algebra.
Area, Volume, Perimeter, Distance [5.0]
This text goes above and beyond the topics expected at grade 7. Area of rectangles, parallelograms and triangles are reviewed early in the course and are only mentioned in the chapter concentrating on areas of "irregular" shapes, circles, "shaded" areas, and the volume and surface area. The later sections include cones, spheres, and irregular three dimensional shapes. Congruence and similarity of polygons are given a very thorough treatment. Similar solids are discussed and the ratios of the measures of similar solids are taught, including the use of scale factors. The Pythagorean theorem (assumed in the "Toolbox" and proven in the previous book in the series) is used to solve right triangles. Right triangle trigonometry is also introduced with practice problems using sins, cosines and tangents (This is high school level material, well beyond pre-algebra).
Program Quality Evaluations
Mathematical Depth [4.7]
This book has first rate mathematical content. Nearly every possible important topic to prepare students for algebra and geometry is covered in depth. 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. They could also be substantially through the content of many geometry courses. To truly take advantage of all that is in this book, students would have to come into the course very well prepared (the course just before this in the series is itself nearly an acceptable pre-algebra book). On the other hand, students who are moderately well prepared after grade six can become quite ready for algebra if the course is appropriately trimmed back to concentrate on the key pre-algebra and pre-geometry topics.
Quality of Presentation [2.8]
With a moderate number of good choices by the teacher this course has a good presentation. On the other hand, there are a number of opportunities for the teacher to make wrong choices. First of all, the book is very long. Realistically, how many classes will cover all 714 pages of text, especially with the necessity of including time for tests, enrichment activities outside the lessons, and all those things that disrupt the flow of a class (assemblies, teacher sick days, etc.)? Thus the book, as presented, tries to be all things to all people by stuffing in much more material than can be done. Yes there is very high content, but there are also discovery lessons and calculator explorations. Figuring how to apportion time among these, and figuring out and deleting the math content that is beyond the appropriate goals for this course, will be a critical job for the teacher. To give an example of the importance of correct choices, graphs of linear equations, an important, appropriate and well presented topic, comes in chapter 13, which begins on page 616. A class that starts at page 1 and just keeps going may learn many things, but probably not graphing of linear equations.
Unfortunately, some of the exploration and calculator lessons fall short of the rest of the text, making teacher choices even more critical. Algebra tiles were mentioned above. Consider two other examples. The "Using a calculator" technology note on page 15, stresses not only calculators, but, in a misguided "problem solving" lesson, stresses the inefficient and non-analytic problem solving method of "Guess, check and revise." This is a twice wrong message to send to students, especially as the problem, how to determine the sides of a cube of given volume, really lends it self to a true understanding of volume, cubes and roots. Lab 9.3, pages 418 and 419, has students plot right triangles with integer length legs on graph paper, then measure the hypotenuse with a ruler and hypothesize the Pythagorean theorem. Unfortunately, the measurement errors in this exercise will be significant and students will, at best, be able to conclude that Pythagorean theorem is close but maybe not right. What makes this particularly sad is that the book that precedes this in the series, written by the same authors (Passport to Mathematics, Book 2, Lab 8.8 page 410-411), actually has students prove the Pythagorean theorem in its discovery activity related to this topic.
Quality of Student Work [3.1]
The work in this book certainly has plenty of problems at a depth appropriate for the mathematics being taught. The overall rating for work is somewhat lower than it might be because of some inappropriate uses of technology and for some problems related to presentation, as noted above.
Overall Program Evaluation
This is a very highly rated book. Content drives this high rating, with some of the concerns about presentation lowering the rating slightly. Appropriate deletion of misdirected activities would bring the presentation rating up. The book would really work well with top level students, making them prepared for truly first rate algebra and geometry instruction. Less well prepared students, even if capable of being ready for algebra in a year, will require a teacher who carefully picks and chooses material from this book so that the key topics are emphasized and extras of all kinds are given "less emphasis."
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