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 contains:Each of the relatively short chapters contains about four lessons and is arranged around a particular mathematical theme
Lessons are generally free of nonsense. There is a clearly defined exposition with new terms defined and highlighted. This is followed by about 1-3 examples building from the exposition, a set of guided practice problems and then a problem set of about 30 problems. Problem sets usually begin with "mechanical" exercises, then word problems. Two lessons per chapter are followed by mixed review and test prep exercises. In addition to the exercises, each chapter contains one "lab activity." Although not intrusive, these are generally built around algebra blocks and other manipulatives. The end of the chapter contains a review/test exercise and a cumulative one page test prep exercise.
The book contains a moderate number of extraneous illustrations (27 pages in pages 101-150 have unnecessary pictures). Most of these fall in the margins of the page, as do a number of side bar discussions on links to other disciplines or things to remember. In some places, including many of the expositions, these side images can distract from the main focus of the text. This problem is much less serious in the actual problem sets or reviews.
Content Area Evaluations
Properties, Order of Operations [4.0]
This is an above average treatment of this topic. Evaluation and order of operation problems use powers, negative integers, decimals and a few fractions. The commutative, associative and distributive properties are presented clearly, although step-by-step "proofs" are not presented. Good emphasis is put on grouping like terms. Two step simplifications in which parentheses are removed and the like terms gathered are also present.
Exponents, squares, roots [2.5]
This book gives a less than adequate treatment of this topic. The emphasis is clearly on powers and roots of whole numbers, although there is a discussion of powers of fractions and negative exponents in the second half of the book. There is no coverage of variable expressions with exponents, nor is there coverage of multiplication or division with powers. How to write a number in scientific notation is taught, but, in the absence of multiplication or division with exponents, it is not applied to problem solving or calculation.
Fractions [3.5]
All of the basic skills of fractions are covered. The text also directs students to the student handbook section at the back of the book for additional instruction and examples. These sections in the handbook are clear, concise, easy to locate, and free of distractions. Answers to the practice problems in these review sections would be helpful. Most of the manipulations with fractions are introduced early in the text, consistent with the fact that there is substantial overlap with material covered at earlier grades. For some reason, some topics involving fractions, such as conversion to repeating decimals, occur 300 pages later. For conversion of fractions to decimals, two techniques are taught and shown in examples. The more universally applicable, divide the numerator by the denominator, is taught second. As a first technique students are taught to see if the denominator is a factor of 10 or 100, then to multiply both numerator and denominator by a number that will convert the denominator to 100, giving a fraction of N/100, which is then easily a decimal. This certainly works, but seems to require a lot of extra time to check whether the denominator is a factor of 100 and what its partner factor is. For many students, this seems to create a "short cuts make long delays" situation. Of course, students should eventually learn to recognize the decimal equivalents of common fractions, this method does not seem reasonable as a primary method. Directions, as part of a regular lesson, of how to punch numbers into a fraction calculator, seem to run counter to the mental skills that are attempting to be taught for fractions and their manipulations. After all, in algebra the skills of fractions will be applied to symbols, making the fraction manipulation skills, not just the answer, important at this level.
Decimals [4.0]
With the exception of converting repeating decimals to fractions, all appropriate topics are covered. The examples are clear and the associated word problems are varied and appropriate.
Percents [4.5]
This topic is covered in two chapters. Chapter 15 focuses on computational methods. Chapter 27 concentrates on percent increase/decrease, discounts, commissions and simple interest, although some problems on some of these topics also appear in chapter 15. Overall the coverage of percents and their application is superior. There are many examples and plenty of independent practice. The various different applications in chapter 27 reasonably combine the mathematics and meaningful real world situations. The level of difficulty is sufficient for advanced students. If there is a weakness in the coverage of percents, it is in the description of how to calculate percent. As discussed in the section on fractions, conversion of fractions to percents is first taught in terms of inferring the number to multiply numerator and denominator by to put the fraction in terms of one hundredths. This seems like the wrong approach to take first. The book then goes on to approach conversion to percent as a proportion problem (e.g. 5/12 = x/100) which is a reasonable way to do things that seems to work well for many students and leads naturally into the decimal times 100 approach to calculating percents. This second approach would be a far better way to begin. Then, through time, students could be expected to recognize/memorize, the percent/decimal equivalents of commonly appearing fractions such as n/2, n/3, n/4, n/5 etc.
Proportions [4.0]
This is an above average treatment of this topic. Proportions are actually introduced in the equation solving chapter, even before the concept of ratio is discussed. Although unconventional, this is a perfectly appropriate place to use cross multiplication to solve for an unknown. Proportions are later used in the percent, similarity and change of scale sections. Ratios are clearly presented with lots of attention paid to unit rates. There is no coverage of conversion between unit systems and only minimal coverage of within systems conversion, essentially all of which comes in the student handbook at the back of the text. The various change of scale and so on problems tend to strip of units within the equations and then add them back on to the answer at the end. Although this is OK some of the time, students should be introduced to unit analysis as a way of verifying that the equation they have written is reasonable and that the answer at least makes sense in terms of the units.
Expressions and Equations - Simplifying and Solving [3.5]
This book provides an above average treatment of equations, although there are some weaknesses. One and two step equations are solved with positive and negative rational numbers, the distributive property is used in simplification as is grouping of like terms. Equations with variables on both sides are omitted. This book makes a serious error by stating that inequalities are preserved by addition, subtraction, multiplication or division of both sides. The cases where this is not true (multiplication or division by negative numbers) is ignored. This could cause serious confusion later on as many students will remember the incorrect "rule" and have to unlearn an error. If inequalities are to be solved using the properties of inequalities, all of the properties must be stated correctly.
Expressions and Equations - Writing [4.0]
This book does a particularly good job of guiding students through the steps or writing an equation. The concept of choosing a variable and then writing all unknowns in terms of that variable is clearly demonstrated in the examples. The students are also write multistep inequalities. There is a particularly good job on the "test score average" problem, in which students must write and solve an inequality to figure out the minimum score necessary to earn a certain average. Numeric and algebraic expressions are presented in a single lesson with some indications of how to translate particular phrases into mathematical expressions. It would be better if there were even more examples and even more practice problems on just this translation.
Graphing [3.0]
This topic is presented relatively early in the course, about one third the way through the book. Domain, range and the vertical line test for functions are discussed, which is unusual in a book at this grade level, but not out of place. Linear equations are defined and graphs are drawn by calculating tables of values and plotting points, but not by any other method. The concept of slope is not presented, nor are x or y intercepts discussed. Multiple equations in the plane are plotted only as parallel lines in a calculator exercise. Since this was not developed into understanding the meaning of the "b" term in y=mx + b, the exercise goes nowhere. One variable inequalities are plotted on the number line, but there is no coverage of inequalities on the plane. Other types of graphs based on data, with a few brief exceptions, are presented late in the course. This is appropriate prioritizing. The early focus should be on graphs arising from equations and how to make, use and interpret them.
Shapes, Objects, Angles, Similarity, Congruence [3.5]
The books presentation of these topics is about average, strong in some areas and weak in others. Too much emphasis is placed on such "extra" topics as transformations, fractals, and tessellations. "Regular" is only used to describe solids, not two dimensional polygons. With the exception of a thorough similarity section, two dimensional objects are given far less attention than three dimensional ones. For instance, various quadrilaterals are defined in the "handbook" but never clearly categorized in, say, Venn diagrams, when the subset relationships should come across. Diagonals and interior angle sums are missing.
The treatment of constructions is excellent, however. The book presents basic constructions such as angle bisectors and perpendiculars, but also uses congruence theorems such as "side angle side" and "side side side" to have students construct triangles. The triangle inequality theorem is indirectly presented in one of the discussions.
Area, Volume, Perimeter, Distance [2.5]
This topic is treated in a less than satisfactory manner. Formulas for areas of parallelograms, triangles, circles and trapezoids are presented clearly but briefly, mostly in the end of the book student handbook. The book is weak on derivation, for instance, in demonstrating how a triangle can be thought of as half a parallelogram. The surface area units have reasonable diagrams, but the volume formulas are preceded by silly activities involving filling containers with popcorn. Better to present the formulas and note that it is possible to determine the formulas exactly by mathematical derivation, deriving those which can be done easily and noting that derivation is beyond the scope of the course for the others. Similarly, a calculator and manipulative exercise is directed at determining if the "experimental" value of pi is equal to the numerical value stored in the calculator. Better to note that the errors in measurement make this "experimental" technique inherently inaccurate, but that we can do this mathematically for a general case. Then perhaps at least show how we could progress to an ever more accurate value, say by going to ever higher regular n-gons as approximations of circles.
The geometry units seem too spread out for a student to be able to put all the pieces together. The order is unusual. For example, diameter and circumference are defined in the late 200s while radius isn't mentioned until the 400s.
The students are not asked to compute irregular areas by breaking up larger shapes into triangles, rectangles and parts of circles. There are a few problems requiring that students find the areas of shaded portions of a figure, but these are hidden in the "geometric probability" unit away from all the rest of area.
The Pythagorean Theorem, like many of the area formulas, is just presented without any justification or even a geometric interpretation.
In summary, the basic pre-7 material is present, as is some grade 7 material, but depth, sequence and instructional method are less than optimal.
Program Quality Evaluations
Mathematical Depth [3.8]
The mathematical depth of most topics is relatively high, but there are some exceptions, for example, exponents, powers and roots are poorly covered. Even within these weaknesses, the mathematical content of this book is such that students could be prepared for algebra having taken this course.
Quality of Presentation [2.6]
In terms of presentation, this book is decidedly mixed. Some things are done better in this book than in most other books while others are badly done. The difficulties in presentation have lowered the subscores in a number of content topics. The particulars of these errors or difficulties are discussed in the individual topic sections.
This book spends relatively little time redoing sixth grade math, thus allowing students to spend their time on the goals of grade 7. Integers are presented relatively early, thus allowing both positive and negative numbers to be used in all later exercises.
The relative large number of chapters and extra activities means that topics at the end of the book may not be covered in the average class. Since some key pre-algebra skills such as percent word problems (chapter 27) and area and volume problems (chapters 24-26) appear very late, it may be critical for the teacher to decide which earlier activities to skip to be sure to give these topics all the time they deserve.
Chapter 28, which is largely related to graphical descriptions of physical situations contains a number of errors. Some of these have to do with misinterpretation of actual physical situations, other have to do with mistranslations between various representations of the data (e.g. a change between distance vs. time and speed vs. time). Luckily, this is a topic that does not need to be covered at this level and can be skipped without detracting from the positive points of the book.
Quality of Student Work [2.7]
The student work generally follows both the positive and negative aspects of the presentation. With appropriate choices it could support relatively efficient learning of the topics at the level of the presentation.
Overall Program Evaluation
As noted above, this rating reflects a slightly higher content level and a mixed and variable level of presentation. The content level of this book places it right in the middle tier of books relative to use as a pre-algebra book. Like most other books in this middle level, it could, based on content, serve well as a pre-pre-algebra book and could serve as a pre-algebra book for some students. On the other hand, the content gaps make this book a less good choice than a solid pre-algebra book for those students who are expected to take algebra in the year following this course.
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