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 Fifth Grade Program Reviews.
Student Text Structure
The student text contains 600 pages divided into 13 chapters:
The introductory material for each chapter begins with a theme which introduces some context that will appear later in the chapter. Then a What do you know? section appears that presents some context problems of mixed varieties, followed by a vocabulary list of terms coming in the chapter and their page numbers.
Each chapter is then divided into 7 to 12 lessons. In each chapter, one of the lessons is a Problem Solving Strategy lesson and one is a Problem Solvers at Work lesson. A number of the lessons in each chapter (2 to 6) will be identified as Explore Activity lessons. These include instructions for students to work in pairs or groups.
In the middle of each chapter, between lessons, there appears a Mid Chapter Review, a Math Connection, and a Real Life Investigation. The connections are designed to develop number sense, spacial sense, technology sense, or, in one case, algebra sense. The investigations could consume multiple class periods if completed.
Nearly all lessons are divided into sections called Learn, Check, and Practice. The Learn section presents the material and frequently includes examples and explanations. However, there are frequently questions posed of the students in this section that remain unanswered by the text. The Check section presents some problems and frequently asks students to explain their work. The Practice section includes the student exercises and typically also some mixed applications. A small Mixed Review/Test Preparation section often appears at the end of the Practice section and gives a few review problems, typically in numeric form without context material.
Cumulative reviews appear after chapters 3, 6, 9, and 13. The back matter of the book contains:
Content Area Evaluations
Multiplication and Division of Whole Numbers [3.2]
Multiplication of Whole Numbers
This topic is part of chapter 4 which also addresses multiplication with decimals. Lesson 1 begins the extension of the multiplication facts to multi-digit cases by introducing multiplication with multiples of 10 (both factors have one significant digit) and by powers of 10, as well as the commutative, associative, identity, and zero product properties. The material is presented with minimal exposition that doesn't really explain the concepts. Students are expected to infer that the process is to multiply the leading digits and transfer all of the zeros to the product from minimal examples and are instructed to Talk It Over. The properties are introduced with a note suggesting that they are useful in mental multiplication. Each of the four properties is listed with a short numeric example without annotation. Four more examples with very minimal annotation follow, and the properties are not well-represented. The Check section contains 12 problems and the Practice section contains 42 problems. Thus, although the quantity of student work is ample, the presentation is weak.
The second lesson is on estimating products by rounding to the largest place before multiplying. Students are clearly instructed to round each factor to its greatest place. More questions are posed of students at the end of the Learn section. Again, practice is ample.
The third lesson is on the distributive property. In the Explore Activity, students are to work cooperatively with counters to model a product and then separate the model into two arrays. Several questions are asked about the process. Then, the distributive property is explained in a way that may not be easy for students to understand. Two examples illustrate the property with pictures of arrays and equations, but without further annotation. As is typical for this text, students are asked questions about the process, and 37 problems follow.
Lesson 4 is on multiplying by 1-digit numbers. The distributive property is cited as the standard algorithm is introduced, but the process is not illustrated with the distributive property being applied in the manner given in the prior lesson. Three steps are presented in the multiplication of a 3-digit number by a 1-digit number with modest annotation. For example, the regrouping annotation is not explained beyond the instruction to Regroup if necessary. Three other examples are worked briefly. The Check section presents 5 numeric problems, 2 of which are triple products. The Practice section presents 31 problems, mostly numeric. A short more to explore section then appears giving exponent notations for repeated multiplications.
The next lesson is a problem solving strategy lesson for multi-step problems. Multiplication is involved in the example. This is followed by a mixed set of application problems. Lesson 6 of the chapter is devoted to multiplication by 2-digit numbers. The topic is introduced with a grid and a reference to the distributive property. Again, however, the notation doesn't represent the distributive property in the normal manner. An example is worked in three steps with limited annotation. Three other examples are presented. Calculators are used in some cases in the examples. The Check section presents 5 numeric problems and some questions requiring explanations. These are followed by 25 numeric problems up to 2 digits by 5 digits and an assortment of other problems involving multiplication by 2 digits. The Midchapter Review follows this lesson and includes multiplication problems.
Thus, whole number multiplication assumes students come in with a knowledge of the multiplication facts and builds through multiplying by 2-digit numbers but not beyond. It is likely that much of this material is review for grade 5 students, and thus the growth of multiplication ability is limited. Explanations are weak, further suggesting that the text expects students to come in with some of this content covered already. Problem counts are generally ample.
Division of Whole Numbers
Division of whole numbers and decimals are treated in chapters 5 and 6. Chapter 5 addresses division by 1-digit divisors. The first lesson introduces division with remainders as dividing into equal groups and illustrates this for dividing 16 by 3 and dividing 16 by 5. Students are to talk about the similarity of these problems. Three symbolic notation systems to represent division are given. The terms dividend, divisor, quotient, and remainder are highlighted and a note refers the student to the glossary. Fact families are introduced as a way to show the relationship between multiplication and division. Three examples are given, but solutions are based on number facts with remainders. The Check section contains 14 simple problems. The Practice section includes 54 problems, but these focus on the division of a 2-digit number by a 1-digit number, again only focusing on facts and remainders.
The next lesson is devoted to mental math based on patterns using powers of 10. As with multiplication, this is illustrated numerically as a pattern across increasing powers of 10 without explanation. The third lesson is on estimated quotients using rounding to form compatible numbers.
Lesson 4 introduces the standard algorithm for division by 1-digit numbers. A model of 94 elements is divided into three equal groups, first putting 3 10's in each group and then 1 unit in each group, with one unit left over. Beneath the steps are the steps worked out using the standard algorithm, but the process is not explained. Another example is worked in three steps using the standard algorithm with only modest explanation, and the calculator is shown as an alternative. Three more examples are shown with no explanation at all, and one final calculator example. The Check section contains 5 numeric problems and some items asking students to answer questions about the process. The practice section includes 61 problems and some application problems. These run from two digits by one digit up to five digits by one digit.
Lesson 5 addresses the situation of finding zeros in the quotient. A worked-out example calls attention to the need to write a zero in the quotient. There are three more examples without explanation and one calculator example, followed by 5 problems in the Check section, and 16 numeric problems and some application problems in the Practice section. A fraction of a page is devoted to the short division algorithm but the process is without explanation. The next lesson is on minding means, so division is involved but not illustrated or explained as a procedure. A connection is devoted to the divisibility rules, and the rules for 3, 6, and 9 are given. The balance of the chapter relates to division with decimals.
Chapter 6 takes up the topic of division by 2-digit divisors. The first lesson expands upon single-digit divisors by taking cases that incorporate powers of 10. The examples are without annotation except that, as is the case for the powers of 10 section for both multiplication and division in this text, the zeros are blue.
The second lesson addresses estimated quotients and refers to compatible numbers, as was done with 1-digit divisors earlier. The third lesson uses an explore activity to model division by two-digit divisors. Place value models are show for division into equal groups with remainders. The examples are limited in scope to small divisors. There are 32 problems in the practice section that are also limited in range. Place value models are allowed in the problem set.
Lesson 4 addresses the standard algorithm. A worked out example has a modest amount of annotation. Two more examples are given with some explanatory notes, one more without explanation, and one solved using a calculator. Placement of the first digit is given by estimation. Special attention is not given to zeros in the quotient. The 5 numeric Check problems run to moderate difficulty for 2-digit divisors, as do the problems in the Practice section.
Before the topic continues, a lesson titled Guess, Test, and Revise appears as a problem-solving strategy. Lesson 6 uses 2-digit divisors with larger dividends. Again, worked-out examples are provided with moderate annotation, and calculators are included in the examples. These are followed by a reasonable quantity and range of 2-digit divisor problems, no mention is made as to whether or not calculator use is permitted.
Thus, this text provides reasonable attention to division through 2-digit divisors. As with multiplication, the text starts at a minimal level and doesn't get as far as desired. Also, the explanations and examples are often weak, and questions to students are typically left unanswered by the text.
Decimal Multiplication and Division [2.9]
Decimal Multiplication
The treatment of multiplication of decimals is partially mixed with the treatment of multiplication for whole numbers. In the first lesson of Chapter 4, multiplication by powers of 10 is addressed, and this includes multiplying a decimal by a power of ten. Explanations here are not really given, students are supposed to talk about the patterns they see. The second lesson is on estimating products for both whole numbers and decimals multiplied by whole numbers. However, the estimates for decimals are accomplished by rounding to whole numbers.
After some work with whole numbers, decimals are discussed again in lesson 7 of the chapter. The topic here is the multiplication of a decimal by a whole number. This is first introduced in a money context. Students are told to multiply as with whole numbers and place the decimal point. Since they are working with dollars and cents, there is no real issue with the placement of the decimal. In the second example (0.6 times 4), the decimal is placed by estimation. The example is worked in standard notation and in a figure of grids. Two more examples are given. The decimal is placed by estimation in the first case. The second shows a picture of a calculator and appears to suggest that the decimal is placed either by estimation or by the calculator, but no explanation is given.
The next lesson introduces the multiplication of two decimals using models with grids and double shading. One example is given. Students work in a group to model 8 more problems and answer questions about the process. Two more models are shown in the text, and the students are asked what rule they can use for placing the decimal point. Thirteen practice problems are given multiplying single-digit decimal factors, with the instruction that graph paper may be used.
The next lesson uses the standard algorithm for multiplication and uses estimation and counting decimal places for the placement of the decimal. Two examples are shown worked out. Arrows point and note one decimal place in each factor and 2 decimal places in the product. A note says You can use the estimate or count decimal places. The second example is also worked by a calculator. The examples do not explicitly say that the sum of the places in the factors is the number of places in the product. Students are to talk about this process.
In the next example one factor has one decimal place and one has two decimal places. Again, arrows and notations indicate the number of places in the factors and the products. The example is also shown worked with a calculator. Two more examples are given, plus one worked with a calculator, and a note indicates that a calculator does not show unnecessary zeros. Nine practice problems are given and two questions asked in a check for understanding section. This is followed by 9 problems for decimal placement and 47 to find products.
The next lesson addresses the need to add leading zeros in the product. The example simply instructs students to write zeros to place the decimal point in the product. Two more examples and one calculator example follow without explanation. This is followed by 34 problems.
In summary, topics in the multiplication of decimals appear in several lessons. This includes two lessons that address the multiplication of decimals by decimals. Most problems are at a low difficulty level. Explanations are not always clear or leave questions unanswered. The student work is generally ample through two significant digits by one significant digit, but not beyond.
Decimal Division
Division of decimals is first addressed in lesson 8 of chapter 5. Division of whole numbers and division of decimals are interwoven, so that both are addressed in this chapter on single-digit divisors. In this lesson, models are used to illustrate division of decimals by a whole number. At first, students are to work in a group to model the division. The task is 3.27 divided by 3. Three whole models, 2 tens (tenths) models, and 7 unit (hundredths) models are to be divided into three equal groups. The process continues for 8 other modeled problems. The text then illustrates the process for 3.42 divided by 3 in four modeled steps. The standard algorithm is shown next to the models, but only minor annotation is given to it.
The next two lessons continue division of decimals by whole numbers. The first of these includes the explicit instruction to place the decimal point in the quotient above the decimal point in the dividend and divide as with whole numbers. This is repeated in a second example. Three more examples are shown without further explanation, two are also shown worked with calculators. The second of these two lessons introduces rounding the quotient in the context of money. An example is shown with annotation. Problems in this section may have 0, 1, or 2 decimal places in the dividend, so they are taken out of the money context. Some of these problems include the explicit instruction to round the quotient.
The text treatment of division continues in chapter 6 with 2-digit divisors. Whole numbers are treated first, and the topic of decimals appears again in lesson 7 which introduces division by powers of 10. Students work in groups to explore this topic by making a table of values and looking for a pattern. Patterns yielding decimal quotients are then given in the book with more questions about the process left for students to answer.
The next lesson continues the topic by dividing decimal values by 2-digit whole numbers using the standard algorithm. The example instructs students to continue dividing by writing zeros in the dividend. More examples and problems follow. The examples also note rounding the quotient to the nearest cent in one case and to the nearest thousandth in another. A small more to explore section at the end of this lesson introduces terminating and repeating decimals very briefly.
In summary, several lessons are devoted to the division of decimal numbers. However, the division of a decimal by a decimal is never addressed. The presentation, as far as it goes, is fair although some questions are left up to students to answer. Calculator use is indicated periodically throughout.
Area of Triangles [2.5]
The area of right triangles is introduced in Chapter 12. In a cooperative exploration, students work with a partner and cut out rectangles and draw diagonals on them. They are to make a table showing the area of the rectangles and triangles, but not given the way to find the area of the triangles (they are supposed to discover that the area of the right triangles is half that of the rectangles). They are asked questions about these relationships.
On the next page, they are given the formula. The terms base and height are highlighted but not defined. The glossary notes that a base is a side of a polygon, usually the one at the bottom. It defines height as the distance from the base to the top of a figure. These definitions are neither very precise or useful. Students then find the area for 8 right triangles presented as figures - six cases where the base and height are given as numbers, and two word problems.
There is no generalization to non-right triangles. The reasons for wanting to know the area of triangles are not given. Thus, the coverage of this topic is very minimal if students even make it this far in the text. Given that a whole lesson is devoted to this topic, one might expect further progress.
Negative Numbers [1.0]
This topic is either not present or appears to such a minimal extent as to be effectively not present.
Powers, Exponents and Scientific Notation [2.6]
Powers are introduced in Chapter 4 in connection with the extension of multiplication by using powers of 10. The values of 10, 100, and 1000 are introduced as decimal powers of 10 without much explanation and without reference to exponents. Thus, this first appearance should not be considered as part of this topic.
Exponents are introduced later in Chapter 4 as a shorter way to represent repeated multiplication without further objectives being defined. A short explanation and 10 problems occupy a total of about 1/3 of a page. An example is given with arrows pointing to the base and the exponent. Translation from repeated multiplications to exponents is practiced in 4 problems. The reverse translation is practiced in 6 problems.
Thus, this is a minimal coverage of this topic, not extending to the case of writing prime factor strings or any justification other than use for shorter notation. This minimal treatment is likely to have little benefit to students.
Program Quality Evaluations
Mathematical Depth [2.8]
The mathematical depth of this program is less than desirable. Student growth to mastery of multiplication and division of whole numbers cannot be expected. Multiplication of decimals could also be further extended. Division of decimals is restricted to whole number divisors. Although area of triangles is presented, it is not covered in the general case. Negative numbers are not covered. The material on powers and exponents is so minimal as to be of little value.
Quality of Presentation [3.2]
The quality of presentation in this program is modest. Perhaps the best aspect is the quality of the examples provided. The objectives of each lesson will not always be clear to the students, and the explanations are often inadequate. Particularly concerning are the situations where important conceptual elements are left open, often as questions for the students, thus risking a lack of closure.
Quality of Student Work [3.2]
The quantity of student work is adequate although sometimes may be inefficient. However, the range of work is quite limited and will supports only limited student achievement.
Overall Program Evaluation
The overall rating for this program is only fair. In general, the level of mathematics supported is below review expectations and student achievement will be consequently be limited. The examples are reasonably clear, and the quantity of student work is adequate but only appears at low achievement levels. There is some concern with the lack of closure in the presentation. Students success would be limited to the fairly low achievement levels supported by this program.
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