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 is divided into 12 chapters:
Each chapter is divided int 9 to 15 lessons. In addition, each chapter includes two Checkpoint sections and one Problem Solving section. At the end of each chapter are found extra practice, a chapter test, a performance assessment, and extension, and a cumulative review.
The introduction to each chapter includes Real Facts and Real People as a brief presentation.
The typical lesson structure is divided into three sections: Learning About It, Try It Out, and Practice. The first section presents the concept or method in ways that vary by topic. The Try It Out sections are usually brief problem sets. The practice section typically presents standard student exercises, some of which are in a problem-solving subsection. There is another subsection called Review and Remember that provides the students an opportunity to rehearse prior material.
The text typically deviates from the regular lesson structure in three lessons in each chapter to provide sections on developing problem-solving skills, a problem-solving strategy, and a problem-solving application. The text also includes from one to three lessons per chapter called Explore that have a slightly different structure.
Content Area Evaluations
Multiplication and Division of Whole Numbers [3.8]
Multiplication of Whole Numbers
Multiplication of whole numbers appears as the focus of chapter 4. The first lesson in this chapter addresses the multiplication properties. The third lesson addresses mental multiplication for patterns of 10, 100, and 1000. Lesson 4 is on estimating products. Multiplying by one-digit numbers is given in lesson 5. Lesson 7 addresses partial products. Lesson 8 introduces multiplying by two-digit numbers. Lesson 9 give more on multiplying by two-digit numbers, and lesson 10 addresses multiplying greater numbers.
In the lesson on one digit numbers, lesson 5, the topic is introduced by the statement, You can use basic facts to multiply by one-digit numbers. A problem situation is introduced. The example shows the standard algorithm with some explanatory notes. It also shows the same problem solved with mental math and makes reference to the Distributive property. Students are to think and discuss the similarities of the two methods. This is followed by 5 problems in the Try It Out sections. Next, 20 numeric problems appear in the Practice section. These range from 2-digits by one-digit to four-digits by one-digit, and include one problem where the 1-digit number is the first factor. These are followed by an application problem requiring multiplication, another requiring estimation, and a journal idea.
This lesson is followed by a problem-solving lesson and a Checkpoint that provides more practice on mental multiplication, estimating products, and multiplying. Then an Explore lesson presents partial products in an array context. Lesson 8 introduces multiplication by two-digit numbers in the context of partial products and multiples of 10. In each example, the solution using an array to represent the partial products and using the standard algorithm are presented. Some discussion of the steps is given, including explicitly noting the use of 0 as a place holder when multiplying by the tens digit. This is followed by 30 numeric problems and an assortment of other problems.
The next lesson addresses multiplying a 3-digit number by a 2-digit number. The example gives the solution by the standard algorithm and with a calculator. In the practice problems the students choose either paper and pencil or a calculator for a solution.
The next lesson extends the skill to the multiplication of two 3-digit numbers. The example again illustrates both the standard algorithm and the calculator. The practice problems again allow students to choose paper and pencil or calculator use.
In summary, the lessons devoted to the multiplication of whole numbers take the student from the single-digit case up through multiplying a 3-digit by a 3-digit number. The explanations are fairly clear, but the text alone would probably not be sufficient for students to learn the standard algorithm. The text does present both the standard algorithm and an alternative in each case, and does devote several lessons to this topic. On the other hand, students are given the option of calculator use for the more difficult problems and could potentially avoid practice of the more difficult problems in this way.
Division of Whole Numbers
Chapter 5 introduces division of whole numbers. This begins with the relationship between multiplication and division and fact families in the first lesson. The third lesson works with multiples of 10 in division to expand the range of mental math. Lesson four covers estimation of quotients. Lesson 5 addresses divisibility and actually gives the divisibility rules for 3, 4, 6, and 9. Lesson 6 provides information on the interpretation of remainders. However, the basis for decisions on what to do with remainders is not given. Finally, in Lesson 7 the division by single-digit numbers is introduced. The first example shows groups of tens and units while the standard algorithm is used to divide 96 into 8 groups of 1 ten and 2 ones. The second example extends this to three-digit dividends. This is followed by numeric problems dividing by single digit numbers and 4 items where students must fill-in missing digits in worked-out problems.
Lesson 8 addresses short division, and the steps of an example are shown, providing students with an alternative for this sort of problem. Lesson 9 focuses on zeros in the quotient and provides more examples and practice with the standard algorithm and single digit divisors. Lesson 10 is a problem solving lesson that does not focus on division, but the next lesson addresses finding the mean which is an application of division. Lesson 12 applies the standard algorithm for division by 1-digit numbers to cases with larger dividends.
Chapter 6 extends the process to 2-digit divisors. As is typical in this text, the first step is to extend the process by using multiples of 10 to get to larger numbers, followed by another lesson on estimating quotients, this time for the 2-digit divisor case. Lesson 3 is a problem solving lesson that does not focus on division. Lesson 4 returns to the use of estimation, this time to get the first digit in the quotient. In so doing, there is a return to the standard algorithm. Lessons 5, 6, and 8 repeat the use of the algorithm with 2-digit divisors with increasingly large dividends. Thus, the method is presented multiple times with increasingly difficult problems.
Decimal Multiplication and Division [3.7]
Decimal Multiplication
Chapter 8 is on both the multiplication and division of decimals. In the first lesson, patterns are studied to learn about multiplying decimals by 10, 100, or 1000. The table shows arrows to indicate movement of the decimal place. Two examples also show this movement. Students are to write a rule for the process and complete three sets of patterns. The problem set then contains 23 problems for finding products and a few application problems.
The next lesson continues multiplication of a decimal by a whole number. The first example illustrates both a grid model and the standard algorithm. The annotation clearly states that students are to place the decimal by finding the total number of decimal places in the factors, even though this is a decimal times a whole number problem. More examples and problems follow. These include applications problems and a journal writing idea.
The third lesson deals with estimated products. Only one factor is a decimal and it is rounded to a whole number in some cases or to a simpler decimal in others. Although the method is to round to the greatest digit, problems focus on rounding to an integer.
Lesson 4 has a problem-solving focus for multi-step problems and decimal multiplication returns in lesson 5. This lesson presents the multiplication of two decimals as models using grids. Students work with partners to double-shade the grid to find the product of 0.3 times 0.5. The steps are explained clearly and a second product to model is assigned. Practice problems involve matching number sentences to models.
Lesson 6 then combines the illustration in a grid with the standard algorithm notation. The annotation notes decimal placement both by estimation and by counting the total number of places in the factors. More examples follow with annotation. Ample problems follow including problem solving applications.
Lesson 7 covers the situation where extra leading zeros may need to be inserted in the product. The explanation is clear and the example is also shown worked by a calculator. Since it is a money context, the solution is also rounded to the nearest cent. Some of the problems that follow may be worked with calculators. A few application problems are included.
In summary, the treatment of multiplication of decimals covers several lessons. The explanations are generally good as are the examples. The problem sets contain a good quantity and range of difficulty levels.
Decimal Division
Lesson 9 in Chapter 8 begins the treatment of the division of decimals. It begins with the case of division by powers of 10. Students are to study the pattern in the table, which indicates with words and arrows that the decimal point is to be moved. Students complete multiplication patterns and mental products in the problem set which includes a few applications and a journal idea.
Lesson 10 presents division of a decimal by a whole number with grids as models. The procedure is illustrated and explained clearly in two examples. Standard notation is given along the way with minor annotation. A few practice problems are to be solved with base 10 blocks.
The next lesson continues the division of a decimal by a whole number. The process is given using the standard algorithm with ample annotation. Students are instructed to place the decimal point in the quotient above the decimal point in the dividend. The first example is in a money context and is also solved using a calculator. Two more examples are not in money context. The first of these notes the need to put a zero in the quotient. The second notes that zeros may have to be added to the end of the dividend to continue dividing. A moderate number of exercises include instructions to check by multiplying. However, a calculator symbol suggests that most may be solved with a calculator.
A few problems in the division of decimals by whole numbers occur off and on in the remainder of the text.
In summary, the treatment of the division of decimals by whole numbers is fairly clear in a small number of lessons. The explanations and examples are reasonably clear. The text does not extend to the division of two decimal numbers.
Area of Triangles [3.6]
This program introduces area of triangles near the end of chapter 11. The chapter begins cutting a triangle off the end of a rectangle and moving it to form a parallelogram, thus showing that the area is unchanged. This leads to the formula for the area of the parallelogram: A = b X h.
The lesson then continues by cutting a parallelogram along a diagonal to yield two triangles with area equal to half of the parallelogram, and thus S = 1/2 X b X h. The problem set then includes 4 triangle figures to solve for area, and 5 more where dimensions are given as numbers, two word problems involving the area of triangles, and a critical thinking corner finding the surface area of a triangular prism. The presentation thus avoids dealing with the area of right triangles as an introductory case, but it also does not deal with heights that are drawn outside of the triangle. The subsequent lesson uses areas of triangles and rectangles to find areas of other polygons.
Negative Numbers [2.6]
The presentation of negative numbers is restricted to an extension (optional) page following chapter 11. In that page, students are introduced to negative numbers in the context of temperatures below zero. Students are then told that negative integers are less than zero, and this is indicated on a number line. Students then represent 4 numbers (2 of which are negative) with symbols. In the remaining 12 problems on this page, students find or label positive and negative points on the number line. No further material is covered. Thus, students are not taught that a negative number results when a larger (positive) number is subtracted from a smaller one. The operations with negative and positive numbers are not introduced, and the use of negative numbers to identify quadrants in the coordinate plane is not covered.
Powers, Exponents and Scientific Notation [2.6]
The only discussion of this topic appears in an extension for chapter 9. This is a page on prime factorization. Writing the factorization in exponent notation is covered. Pointers note the exponent and base in the notation, and examples indicate writing squares and cubes, but the explanation is not thorough. Problems involve exponents for factors up to the 4th power. Students are asked to explain a 4th power and to find 10 to the 1st, 2nd, 3rd, and 4th powers. The entire topic occupies about 1/2 of a page. This is a very brief introduction to this topic that may be of little benefit to most students.
Program Quality Evaluations
Mathematical Depth [3.5]
This program achieves a moderate level of mathematical depth overall, but the development in each topic area is variable. Multiplication of whole numbers is well-supported to the 3-digit case, although calculator use may be a bit excessive. Division of whole numbers is also well-supported but only to the 2-digit divisor case. Multiplication of decimals is addressed with generally clear explanations and examples. Division of decimals by whole numbers is fairly clear in a small number of lessons, but does not extend to the division of two decimal numbers. The area of triangles is treated in a manner that avoids introduction of the limited case of right triangles. The treatment of negative numbers is limited to introductory material, as is the treatment of powers and exponents. Thus, this program extends to some of the more advanced topics, but their coverage is limited.
Quality of Presentation [3.6]
The quality of presentation in this program is adequate and generally consistent across the lessons. The learning objectives are usually clear, as are the examples and explanations. The use of models is a bit excessive for this grade level, but careful instruction could assure attention to the numeric and application problems in most cases.
Quality of Student Work [3.5]
Student work in this program is of adequate quantity. Both numeric cases and application problems are well-represented, and periodic review is included. The range of student work is somewhat limited as more moderate and difficult problems would be helpful.
Overall Program Evaluation
This program provides a fairly clear presentation and a fair degree of student work that should support achievement to moderate levels. In part, however, the ratings for this program benefit from at least some coverage of more advanced topics - area of triangles, negative numbers, and powers and exponents. While the presentation with respect to area of triangles is fair, the other two topics are really covered in a minimal way so that little progress will be made in these content areas. Thus, although some treatment of more advanced material is offered, the program will support achievement only to moderate levels.
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