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 and is divided into 6 units:
There are a total of 157 lessons distributed across these units. It should be noted that the unit classifications are not as strict as they might appear. For example, some of the lessons in Unit 6 are:
Each unit begins with an introductory page that elaborates some of the content areas for the unit as bulleted points. Each unit contains a Mid-Unit Review, a unit test, and a Wrap-Up Project (not included in the total lesson count). Each unit also includes a unit review and a unit practice section (included in the total lesson count).
The back matter of the text includes a cumulative review, a table of measures, a glossary, and an index.
Of the total lesson count, 43 lessons are designated as Applications. Most of the student work in each of these lessons relates to a particular application context. Seven lessons are designated as Revisiting. These are not the review lessons or practices lessons for the units, but tend to review learning that students should have mastered considerably earlier. The Angles and Scale lesson and the Exponents lesson are designated as Preteaching lessons.
Pages within the text are coded at the top with one of the following terms:
A variety of these page types may appear within a single lesson. Also, various icons within the body of the pages may appear, such as:
Some of these identifiers will be noted in this report to give a flavor for the student text.
Content Area Evaluations
Multiplication and Division of Whole Numbers [4.2]
Multiplication of Whole Numbers
Multiplication of whole numbers is addressed in the first unit of the grade 5 text in four lessons.
Lesson 9 extends number facts to calculating products mentally when both factors have non-zero digits only in their greatest place. The lesson contains a practice page and a mixed practice page. Multiplying by 10 or 100 or 1000 is addressed first, then multiplying two numbers with one significant digit each.
Lessons 10 and 11 carry the work for multiplication of whole numbers, and will be discussed in some detail as the approach is somewhat unique. Lesson 13 presents some application problems involving multiplication.
Lesson 10 addresses the standard multiplication algorithm in two problem solving pages. Students read about the context of a rectangular array of floor tiles (not pictured). The text explains that the total by repeated addition or by multiplication. The character in the text goes on to explain the partial products and adds them to get the solution to a 2-digit by 1-digit problem. Each partial product is written under the multiplication problem and summed, so this is not yet the standard algorithm. This is followed by 18 problems with the instruction to use any method the student wishes. Then, the text goes through a 3-digit by 1-digit problem in a similar manner, with three partial products written. Then, Melissa discovered she didn't have to write all the numbers on her paper. After she multiplied 8 times 7, she could write the 6 and remember that she still had 5 tens to add. Thus, the standard algorithm is presented in which the student doesn't mark the carries but simply remembers them. Six problems follow. Then, the lesson continues on a practice page. The Algorithm is presented with step by step instructions for a 3-digit by 1-digit problem. Again, the carried values are to be remembered. The solution is checked by estimation, followed by 20 problems. Finally, the lesson presents a mixed practice page. Most of the page is devoted to a cooperative learning game. The math focus is clearly stated as Adding, multiplying one- and two-digit numbers by one-digit numbers, and mathematical reasoning. Finally, a math journal entry appears.
Lesson 11 extends whole number multiplication to the general case. In a problem solving page students are introduced to an array of partial products in the form of area divided on each axis into the expanded form of the number in question. The student is asked to identify the four partial products (term not used) and their sum. On the next, practice page, the answer is worked out step by step using the standard algorithm with remembered carries for a 4-digit by 4-digit problem. The answer is checked by estimations. Then, 5 problems each 4-digits by 4-digits are given. In mixed practice the students are given 28 numeric problems, but none as difficult as the prior work, followed by two, multi-part application problems involving multiplications and other operations. Another game finishes the lesson.
Thus, in two lessons the text covers the procedures for whole number multiplication. If students come into grade 5 with good multiplication skills, this material is sufficient. However, if students need to learn multi-digit multiplication from scratch, this material would not be sufficient. The grade 4 text from this series provides the needed background. Students coming from other programs would have to be evaluated since more attention may be required.
This text comes close to finishing the topic of whole number multiplication at this grade, and does so in a short time period. This frees students for other learning. Unfortunately, the presentation doesn't relate properties of numbers to the process.
Division of Whole Numbers
The text addresses the division of whole numbers by 1-digit divisors in Unit 1. Division by 2-digit and three-digit divisors is covered at the beginning of Unit 4. Meanwhile, students have completed some of the coverage on decimals and fractions, so these topics will enter the presentation when dealing with whole number division. Lesson 14 uses a problem solving approach to introduce the different treatments of remainders - rounding up or down vs just identifying the remainder depending on the situation. Lesson 15 introduces dividing by a 1-digit division. In this case, the text uses short division. The remainders to be regrouped are to be written into the problem by the student, not just remembered. This is followed by 40 numeric and 4 application problems. Lesson 16 is practice for the short division skill in the context of a game, 12 missing digit problems, 4 application problems, and a cooperative Thinking Story.
In Unit 2 students extend the short division algorithm to yield a decimal quotient and, in a few cases, to divide a decimal number by a whole number. This is followed by a further treatment of remainders expressed as decimal quotients and fractions. In a problem solving context, students now work with five methods: rounding up to a whole number, rounding down to a whole number, expressing the result to a given number of decimal places, expressing the quotient as a mixed number, or giving the result with a remainder.
The standard long division algorithm is introduced in two lessons at the start of Unit 4. The first uses an explore method to introduce division by 2-digit divisors. The text tells a story about dividing $5783 equally among the students in a class of 23. As the students work through the problem in the story they keep a record that looks just like the standard algorithm. The story mentions exchanging $1000 bills for $100 bills and the other needed exchanges. However, rather than describing the technique in the class record in detail, many of the aspects are left as questions - How many $10 bills will they have now? Or, How many dimes were left? In all, there are 13 questions in this two-page explore activity. The answers may not be forthcoming in the subsequent text. An example follows, but explanatory notes like Subtract and bring down the 8 may not be sufficient for students to understand the process. An assortment of numeric and application problems follows. The next lesson addresses division by three-digit divisors. One example is given, but the steps are not explained. Numeric and application problems follow. Division plays an important part in the immediately following lessons which contain various application problems and some numeric division problems.
Thus, the treatment consists of three lessons on remainders, one lesson on short division with a practice lesson, and two lessons on long division with applications following. The students are expected to be able to divide by 3-digit divisors and treat remainders in various ways. The presentation of the standard algorithm, however, is weak in that students would have a difficult time learning it from the text. On the other hand, if students have experienced the algorithms before the emphasis on the treatment of remainders and the applications of division will be useful extensions.
Decimal Multiplication and Division [4.1]
The topics of multiplication and division of decimals are interwoven in this program in that it alternates between them over the course of the text more than in most programs. Therefore, they are addressed together in this case.
Decimals are introduced early in the text, and multiplying and dividing decimals is first addressed in Unit 1. Lesson 22 addresses the multiplication and division of decimals by powers of 10. The procedure is introduced in a few sentences. Examples and problems cover multiplying by 10 first, then by 100, and finally by numbers up to 1,000,000. Division is introduced in a similar manner. Finally, a problem set incorporates both multiplications and divisions.
After a few lessons, multiplication of decimals by whole numbers appears. The example introduces a money context. Solution by repeated addition or by multiplying hundreds of cents are noted as potential methods. Then, the whole number multiplication and the decimal multiplication are illustrated side by side and the similarity and difference noted briefly. This is a useful method for extending whole number operations to the decimal cases.
Students are given a journal opportunity to figure out the simple rule, but the rule is then explained on the next page providing closure. Multiple examples with clear annotation follow, and then an ample problem set is given. These include application problems. The next lesson includes still more problems. Decimal multiplication by whole numbers appears in some problems in the following lessons.
The text returns to decimal divisions at the beginning of Unit 2. The division of a whole number by a whole number to yield a decimal quotient is set in the context of a story where the character happens to use the short division algorithm to solve the problem. This sort of presentation comes across as contrived, probably even to fifth grade students. The example does include the instruction to append a decimal point and a trailing zero to the dividend, and the problem is solved with annotation. Three more examples follow together with instructions to check the solutions. This is followed by a large number of problems with single digit divisors and a large number of application problems. Importantly, examples and problems include both division of whole numbers to yield decimal quotients and division of decimals by whole numbers.
Repeating decimals are not addressed, but having to add multiple trailing zeros to the quotients is noted. In a thorough design, the application problems are designed to bring out five treatments of remainders: round up, round down, express as a decimal, express as a fraction, or report with a remainder. Students are asked to create a problem that would require each of these solution methods. The next lesson continues the use and interpretation of (rounded) decimal quotients.
Multiplication and division of decimals appears next in a lesson in Unit 3. This lesson begins by reviewing the procedure for multiplying and dividing by powers of 10 that was introduced in Unit 1. Then, a situation is introduced for 1200 divided by 200 and, since the units happen to be centimeters, the problem is simplified to dividing 12 by 2. With further elaboration of this problem, students are asked why this is the case. The next page explains that you can divide the divisor and the dividend by the same amount first. Again, the question is posed for students to ponder, but the clear answer awaits them. Four examples are given followed by numerous problems. Most of these are whole number divisions without remainders, although some will have remainders or yield decimal quotients. Many application problems follow.
Multiplication and division of decimals by whole numbers is reviewed in a mixed practice lesson in Unit 6. Problem solving applications follow. To this point, however, multiplication and division of decimals by decimals has not been addressed.
Lesson 144 addresses estimates of decimal products. Then, multiplication of two decimals appears in lesson 145. The correct solution is given with fractions first. Then the answer is computed with the standard multiplication algorithm and counting the places in the factors for the placement of the decimal point in the solution. The explanation of the process is clear and two examples are annotated to illustrate the process. This is followed by 20 numeric problems and two application problems. However, the lesson does not extend to cover the need to supply extra zeros in the products.
Division of decimals appears again near the end of the text. Part of lesson 153 introduces multiplying the dividend and divisor by the same constant to simplify a decimal division problem. The process is extended to multiplying or dividing the dividend and divisor by a constant. A large number of problems appears. Other approaches to decimal placement in the division of two decimals are not addressed.
In summary, several lessons are devoted to the multiplication and division of decimals. These topics appear in sections scattered throughout the course. The program covers multiplication up through the case of multiplying two decimals. It covers division of a decimal by a whole number. It treats the division of two decimals by multiplying or dividing the divisor and dividend by the same constant.
Area of Triangles [3.8]
The area of triangles is introduced near the end of unit 5. In the lesson, students are instructed to draw two congruent triangles (they can trace), but them together to form a rectangle, and find the area of the rectangle. Then, they are told that the area of each triangle is one half the area of the rectangle.
Next base and height of triangles are identified. We choose one side of the triangle as the base. The height is the length of the line segment drawn perpendicular to the base from the opposite vertex. This is illustrated with several drawings, including the height along the edge of right triangles and the height both inside and outside of other triangles. The glossary defines base as a side, not as the bottom. Four right triangles are given as problems.
The next section generalizes further to non-right triangles by illustrating the area of a parallelogram formed by two congruent triangles. The formula for the area of a triangle is then given in two forms. Several problems are presented. However, sample applications requiring the area of triangles are not given.
Negative Numbers [3.7]
Negative numbers are introduced in unit 2. First, in the context of temperature it is explained that if the temperature drops 20 degrees from a start of 15 degrees, the result will be negative 5 degrees. Students calculate a few problems of this sort in the context of temperature and elevation. This is followed by 28 problems in addition and subtraction that may have negative answers. This includes addition where one of the addends is negative, but not subtraction of negative numbers.
The use of the +/- key on calculators is introduced and the same problems are then solved with a calculator.
In this way, negative numbers can then appear subsequently in this program. For example, later in the same unit students are introduced to graphing in the four-quadrant coordinate grid. Students plot points and report coordinates of points in all four quadrants. A function with negative values (negative profit in a business) also appears, and negative numbers show up in several other function contexts. Further topics, such as multiplication and division with negative numbers, are not covered.
Powers, Exponents and Scientific Notation [3.6]
Exponents are introduced near the end of the text in the context of repeated multiplications of the same factor. Several topics are covered in a single lesson.
First, the notation and meaning are explained, as are the terms exponent and base. The explanation to this extent is clear. Four examples and four problems are given in translation from repeated multiplication to exponent notation. No problems are given in this section for translating in the reverse direction.
Use of exponents with a calculator is given next, as the yx key is introduced as one example that lists keystrokes. Students then complete 21 problems finding value through the use of the exponent key. Of these, 9 are simply find the value of an integer raised to an integer power. However, 12 of these problems are products such as 82 * 33 or 38 * 34. No explanation of these is provided, and students will be very unlikely to grasp the concept of precedence for exponentiation over multiplication or the rules of exponents that might apply when the bases are the same.
A table is then given showing the powers of 7 from 1 to 12 expressed in exponential notation and standard notation. Students are instructed to use these facts to solve 16 problems in multiplication and write the answer in standard form. They are not to use a calculator. The first 8 problems are the products of two powers of 7 written in exponential form. The second 8 problems are the products of two powers of 7 written in standard form. Students are using an exponent law here with no explanation.
The lesson ends with an exploration of the ability of the calculator to display large numbers. Students are to enter increasing powers and are asked when the display changes and how it changes. No explanations are offered.
This is a rather unusual introduction to exponents in that it touches on some rather advanced topics, but they are not explained and students will be unlikely to learn them. On the other hand, the experience seems to be sufficient to convey the meaning of exponents and illustrate some uses of them.
Program Quality Evaluations
Mathematical Depth [4.0]
This program received the highest rating for Mathematical Depth of the programs reviewed. Most of the contents sought in the review were addressed reasonably well. Multi-digit multiplication and division, for example, are treated in a way that suggests mastery of these topic areas in this grade level. In part, this may be accomplished by having somewhat higher expectations for students coming into the fifth grade than other programs assume. The coverage of multiplication and division of decimal numbers is likewise at a fairly high level, and includes one approach to understanding the division of decimals by decimals.
The program rating for Mathematical Depth also benefits from fairly good coverage in the more advanced topic areas. Area of triangles is treated in a reasonable manner, except that some applications for this content area would be useful. Negative numbers are covered not only in concept but also in addition and subtraction operations. The introduction to exponents touches on some rather advanced topics that are not likely to be mastered, but the student experience seems sufficient to convey the meaning of exponents and illustrate some of their uses.
Quality of Presentation [3.9]
The presentation is atypically efficient in that it avoids spending too much time on low-level material that should be acquired previously, and this is perhaps a main strength of this program. The explanations and examples are generally adequate, although there are instances when they could be both more detailed and more explicit. Clearer statements of learning objectives would be helpful. The program includes a large quantity of supplemental and support materials as well as a lot of activities that are possible in the classroom. While this adds flexibility for teachers, it could mean that the program is somewhat difficult for less experienced teachers to administer.
Quality of Student Work [4.3]
Quantity, quality, and range of student work is generally good. As with the presentation in general, there are opportunities for varied student work. This means that maximizing student achievement may require careful consideration by the teacher.
Overall Program Evaluation
This program received the highest overall rating of the fifth grade programs in this review. This was accomplished largely on the basis of the mathematical depth supported by the program. In general, the great majority of the review expectations for the topic areas reviewed were addressed. Thus, this was the only fifth-grade program reviewed that could be seen as supporting high levels of achievement. To reach these levels may require careful implementation by teachers, but at least the potential is provided by this program.
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