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 organization includes:Symbols, formulas, and measurement conversion tables are found on the back endpaper and cover. The table of measurement conversion only deals with conversion within a system (e.g. kilometers to meters) and not between systems (e.g. from meters to feet).
Each of the thirteen chapters is given a non-mathematical theme, but the chapters themselves are reasonably focused on related mathematical objectives. These topics, as listed in the contents are
Within each individual lesson or chapter, there are exploratory, cooperative learning or manipulative-using exercises, but most of the actual lessons are focused on a single defined learning objective and contain a clear description of what the student should learn and efficient ways to solve particular kinds of problems. There are generally sufficient examples of an appropriate difficulty level. The examples contain sufficient explanatory material that the logic of procedures is clear, even if formal proof or the use of mathematical properties may not be used in every lesson. Examples are followed with a set of thought questions and guided practice to be done in class. Each lesson contains a reasonable number of problems, each problem set appears to contain word problems and computational problems related to the lesson as well as review problems from earlier in the book.
The book contains a moderate number of extraneous illustrations (30 pages in pages 101-150 have unnecessary pictures), but the placement and size of the pictures generally keeps them from making the lesson pages distracting or too busy.
Content Area Evaluations
Properties, Order of Operations [3.0]
These topics are given average treatment in this text. The instruction on properties and order of operations comes prior to the introduction of negative numbers, so this instruction does not directly deal with negatives. The distributive property should be introduced earlier in the text and used for simplifying expressions. As it is, the distributive property does not appear until the fractions chapter. Earlier use of the distributive property would allow a greater emphasis on and practice of simplifying expressions with parentheses, and would expand the context for this skill. Although the properties are presented, they are generally not used for justification in simplifications.
Exponents, squares, roots [2.0]
This is a less than adequate treatment of this topic. There is coverage of raising numbers to whole powers, and simple roots of numbers, but there are serious gaps. Negative exponents are missing, and the rules for multiplication and division with exponents appear to be lacking even for whole numbers, to say nothing about operations with fractions raised to powers or simplification or evaluation of monomials containing positive or negative whole number powers. Although scientific notation is defined and conversion of decimals with positive exponents to scientific notation is practiced, operations with scientific notation are not presented and the lessons on scientific notation do not lead any farther than the one day's homework. The teaching methods for the parts that appear are reasonable, and should lead to student mastery.
Fractions [4.5]
This is an excellent treatment of fractions. Numerical skills are emphasized, but comparable algebraic applications are brought in at each step. Even the properties are defined with both number and variable symbols. Properties are emphasized in this unit. The conversion of fractions to percents is clearly explained using proportions. The problems include mixed numerals which add rigor to the procedure.
Decimals [4.0]
This is an above average treatment of decimals. All important subtopics are covered, with the exception of converting repeating decimals to fractions and operations with negative decimals. Manipulations are thoroughly covered with clear examples. A little too much use is made of calculators, but the paper and pencil examples are fine. The metric system is presented as part of decimals with sufficient practice. Conversion between decimals and percents is well done. On a negative note, the grid paper models are silly and unnecessary.
Percents [5.0]
This is an excellent coverage of this topic. All important topics are covered and the material is clear. 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.
There are a few small worries. The topic is spread through three non-adjacent chapters (4, 8, 11) which may break up necessary continuity. Some of the use of calculators may be inappropriate depending how much emphasis the teacher places on the calculators.
Proportions [4.0]
This book contains a well above average treatment of ratios, rates, proportions and related word problems. This positive evaluation considers both the quality of the of explanations and of word problems. There are a few problems. The cooperative learning exercises are not necessary efficient uses of class time. There may be too much use of calculators. There is no coverage of conversion between measuring systems. Finally, the treatment of units in problems is not optimal. For example, the book sets up unit conversion proportions or scaling proportions in a way that guarantees that the final answer ends up without units. This will not serve the students well when they get into chemistry or physics classes.
Expressions and Equations - Simplifying and Solving [3.0]
One and two step linear equations are handled adequately, as are simple algebraic expressions. There are no operations involving negative powers. There is some mention of inequalities, but there is a dangerous error in the presentation. After learning to solve equalities, students are advised that inequalities and equalities are solved "in a similar manner," with no mention of the need to reverse the inequality symbol when multiplying or dividing both sides by a negative. This is a dangerous omission as students will be confused when it comes time to learn the proper technique. In some cases, graphing calculators are used inappropriately. For example, on page 16, students are given graphing calculators as an alternative for evaluating 8m when m=6.
Expressions and Equations - Writing [3.0]
Expression and equation writing is presented clearly in a single unit. Except for inequalities, especially as they involve negatives, the appropriate topics are covered. The students are taught to "translate" into mathematical symbols and to use "let" statements (let a = the number of apples) to define variables. Although good on "translation" skills, the book falls a little short in practicing the final application as there are perhaps not enough problems, especially word problems. Otherwise the treatment is algebraically sound.
Graphing [3.0]
This is merely an average treatment of graphing. Real life functions are clearly presented and a section is devoted to linear functions. The students graph these by filling in tables and graphing ordered pairs. Neither slope nor intercepts are covered, nor is graphing of linear equations beyond plotting points from a table of values. There is no graphing of inequalities or of higher degree polynomials by plotting points or otherwise.
Shapes, Objects, Angles, Similarity, Congruence [4.0]
These topics are covered in a number of sections through the text. By the end of the course most of the essentials have been introduced. The sum formula for central angles of polygons is derived in a lab activity before being used for problem solving. The formula for the number of diagonals of an "n-gon" is derived in a sequence of activities. A few constructions are demonstrated although others such as altitudes, medians, angle bisectors and perpendicular bisectors are not covered or not easily found. The level of difficulty and presentation in the this text is fine, but the sporadic placement of topics may be confusing to students. Moreover, there is too much time devoted to tessellation, fractals, and the like.
Area, Volume, Perimeter, Distance [4.0]
With the exception of excessive emphasis on calculators, these topics are nicely presented with sufficient rigor. The section on irregular shapes focuses on "curved" figures requiring estimation as opposed to the type that can be broken into triangles, rectangles and defined parts of circles, and the area accurately calculated. More of the latter type would better build and test student understanding and skills. With regard to calculators, it is absurd that students should be using calculators to multiply 30 x 22 or 12 x 8, or to add 112 and 96 (page 31).
Program Quality Evaluations
Mathematical Depth [3.6]
This book comes very close to having acceptable pre-algebra content, although the book is not necessarily written as a pre-algebra book. It is strongest on the related topics of fractions, decimals, percents and proportions, as well as the geometry related topics dealing with shapes and similarity and area and volume. It is weakest in exponents and scientific notation, falling below even a moderate level of learning in this topic. Other topics are covered at a moderate, but not distinguished, level.
Quality of Presentation [2.8]
In the heart of the lessons, the presentations are generally straight forward and clear, with appropriate examples and guided practice, but there are possibilities that the course can become less focused and distracted from the key content. In many places calculators are used inappropriately and some lessons become lessons in key stroke order. Many of the between lesson exercises could take substantial time away from essential content with relatively little learning benefit. These include many of the "labs" or "group learning" exercises. On the other hand, some, like the lab 9-1B on constructions, are worth the time and attention. Thus, it becomes critical for teachers to judge which of these labs are valuable and which are not. This takes on a particular importance as there is much material in the book and the important chapter on applying percents is third from the end and the chapter on surface area and volume is next to last. Too much time wasted on inefficient group discovery activities early on may mean not reaching these chapters.
Quality of Student Work [2.9]
With the exception of the problem sets or activities containing unnecessary calculator use, and some problems on modeling with manipulatives, the level of work is consistent with students mastering the material in the book.
Overall Program Evaluation
This book has much to recommend it as a pre-pre-algebra book. The content and pedagogy, with appropriate elimination of less effective activities, would support relatively high levels of learning in preparation for pre-algebra. Indeed, a student using this book and then having a serious coverage of the pre-algebra book from the same series and publisher, would have a superior preparation for algebra. The overall score for this book, and the strength in fractions, decimals, percents and proportions, suggests that it might function as a pre-algebra book as well. Although a student could succeed in algebra after using this book, it is difficult to recommend this book for that purpose as there are other better pre-algebra books available, including one from the same publisher.
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