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 28 chapters:
Each chapter begins with a Problem-Solving Activity. Then, each chapter is divided into 4 to 6 lessons. One of the lessons in each chapter is divided into two parts, with the second part addressing a problem-solving strategy. In most chapters, one of the lessons is a Hands-On lesson. In addition, each chapter includes 2 to 5 (usually 3) Mixed Review and Test Prep sections. Each chapter closes with a Review/Test page and a Test Prep page.
Scattered every few chapters is a Checkpoint. Each of the eight Checkpoint areas include sections called Math Fun, Technology, Study Guide and Review, Performance Assessment, and Cumulative Review.
The back matter of the text occupies 150 pages and includes:
The typical lesson begins with a short statement of the objective, however this is usually not useful as it is cast in terms of an application as in Why learn this? You will know how many albums you need for your sports cards.
This is typically followed by exposition that sets up a problem and presents the solution. More explicit instruction may appear in a model or example section. Questions of students are often asked at this point. A short introductory set of exercises appears in a Check section followed by more exercises in a Practice section. The practice may often include mixed problem solving applications. These are often followed by the mixed review and test prep sections which address prior material.
Content Area Evaluations
Multiplication and Division of Whole Numbers [3.9]
Multiplication of Whole Numbers
Multiplication of whole numbers appears as the focus of chapters 5 and 6.
The problem solving activity introducing Chapter 5 is typical for this text. Students are instructed to record their pulse rate at rest, after walking, and after a short workout. They are to make a chart showing their pulse rates. From a side bar, the student is instructed to count beats for 20 seconds and multiply by 3. Students that actually do this activity could spend a class period or a homework night on this activity, multiplying a 2-digit number by 3 a total of three times.
The first lesson in this chapter addresses the multiplication properties. The second lesson is called Recording Multiplication. It uses pictures of counters colored differently to represent different places. The counters are used to show what happens to the place value positions in writing the steps of the standard multiplication algorithm when multiplying by a single digit number. The third lesson places multiplication in the context of finding area, but is largely a repetition of the steps in multiplying by a one-digit number. The remaining lessons in this chapter address simple multiplication problems in the context of area and volume.
Chapter 6 continues the topic. First, the distributive property of multiplication is introduced with rectangles drawn on grids. The second lesson introduces multiplying by two-digit numbers. Three steps are given with some brief notes. Three examples are given worked out but without explanations. While considerable attention was given to the steps in single-digit multiplication, less is given here so that the instruction may not be sufficient. Questions about the process are asked of the student, but not answered by the text in the Check section. Numeric practice consists of 18 problems up to 3 digits by 2 digits, but the highest 3-digit factor is 131.
The next lesson deals with estimating products by rounding each value to the greatest place before multiplying. Lesson 4 addresses multiplying by 3-digit numbers. Four examples are given worked out by the standard algorithm. One of them is also solved using a calculator. About 15 3-digit by 3-digit problems are given in the Check section , and more appear in the Practice section. The student instructions for these problems do not indicate whether or not a calculator can be used. More practice with multiplication appears in the next lesson which gives more area problems.
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 may not be sufficient for students to learn the standard algorithm. The text does devote several lessons to this topic.
Division of Whole Numbers
Chapter 7 addresses division by one-digit numbers. The first lesson addresses divisibility as a concept and some simple cases, not to be confused with the more elaborate rules for divisibility. Lesson 2 addresses Placing the First Digit in using the standard algorithm, relying on divisibility from the first lesson. The place value of some digits is given in the examples. In the process, the standard algorithm is being illustrated in simple cases, but the explanations of the algorithm are meager. Lessons 3 and 4 give more practice in division by single-digit numbers. Lesson 3 focuses on zeros in division problems. Both lessons give additional worked out examples which are important if students are going to master the algorithm. Part 1 of Lesson 5 discusses what to do with the remainder based on the problem situation. Examples are given where the remainder is not used, where it is expressed as a fraction, and where rounding up to the next whole number are required.
Chapter 8 extends the range of division. First, multiples of 10 are addressed, followed by estimation. In the third lesson, attention is then given to placing the first digit in the quotient, which again is the use of the standard algorithm. Lesson 4 has to do with estimation and the quotient digits. An example is worked with a single-digit quotient. Lesson five rehearses the standard algorithm with two-digit divisors. One example is worked with short phrases to explain the steps. Three more are worked without explanation. The quotients are 2 or 3 digits in these examples. A moderate number of student exercises is given covering a range of difficulty in dividing by 2-digit numbers. Larger divisors are not addressed.
Thus, division is covered through 2-digit divisors. There is some treatment of remainders and of estimation in division. However, the development of mastery for large divisors is clearly lacking.
Decimal Multiplication and Division [2.8]
Decimal Multiplication
Multiplication of decimals begins in Chapter 12 of the 28 chapters in this text. The chapter begins with a problem solving activity where items are selected for purchase not to exceed $100.00. Students are to select sports equipment to match the sports the class likes to play, find the costs of these items, and generate a budget. If this activity were done, it could easily consume a day's math time period or more. Multiplication may or may not be used.
In the first lesson, students begin multiplication of a decimal by a whole number by cutting decimal models out of pieces of paper. They model 2 X 0.9 by making two 0.9 models and then are instructed to combine the shaded areas of the two models. The illustrated result shows a whole model and a 0.8 model. Students are then instructed to make models for three pairs of products, such as the pair 3 X 0.5 and 3 X 0.50. They are instructed to describe and explain in writing how to make models to find products. The practice problems that follow continue the use of models.
In the second lesson for this topic, students are to study models for ones, tenths, and hundredths multiplied by 10, 20, and 25. They are given a table of solutions and asked how they can use the pattern they see to place a decimal point in a product. Questions and more models follow in the practice problems. Students are also asked to use mental math to solve these problems.
In the third lesson on this topic, models continue in the presentation for 0.5 of 0.5. An array is shaded twice in three steps, and the annotation notes that the area in which the shading overlaps shows the product. Questions and more models follow. Problems for multiplying single digit decimals then follow (e.g. 0.4 X 0.6) without instructions on whether or not models should be used. Part two of the third lesson is a problem-solving strategy which discusses 0.2 X 0.5 with models. Questions and more models follow.
Lesson 4 also has two parts on placing the decimal point in multiplication. The first part gives a typical objective statement for this text: Why learn this? You can decide whether your answers make sense when you are measuring the mass of an object, such as a pumpkin. Students are to estimate the product of two decimals. The first case is 4.8 X 0.9. Students are asked whether the result is 4.32 or 43.2. They are asked what strategy they used to place the decimal point. An example is presented, 0.6 X 26. Three steps are outlined, basically saying to multiply as with whole numbers and use the estimate to place the decimal. Two more examples using estimation to place the decimal are then given. Finally, a table with four decimal products is shown and students are instructed to look for a pattern. Students use estimation or estimation and patterns to place the decimals in the problem set.
Part 2 of this lesson continues the topic by including information about adding leading zeros to complete the solution. Two examples are given in which the number of decimal places is simply asserted, and thus zeros are added. Students are asked but never told how the number of decimal places is known. This part adds a small section on calculators so that students will investigate the calculator dropping trailing zeros in decimal products.
Lesson 5 introduces the product of two mixed decimals. Two examples are given. In the first, students place the decimal by estimation. In the second, students are finally told to use estimation or count the number of decimal places to place the decimal point in the product. Students are asked if they see the relationship between the total number of decimal places in the factors and the number of places in the product.
Thus, multiplication of decimal values is covered in 5 lessons, with two of these being two-part lessons. While the basic operations are addressed in this series of lessons, the explanations and examples are not very clear. The problem count is minimal and too much time is expended working with models of the operation.
Decimal Division
The division of decimals is addressed in chapter 13 in five lessons. The chapter begins with a problem solving activity. A recipe is generated and students decide how many it will serve. Students are to decide how much they would charge and what the profit will be. A written report is requested. If this project is actually completed, it would consume more than one day's math time. Division may or may not be used, and a calculator is listed among the needed items.
The first lesson asks students to look for patterns in 1000, 100, 10, and 1 divided by 5. Their attention is called to the fact that 1 divided by 5 results in a quotient that is a decimal. Two similar examples follow, and students are instructed to talk about the patterns. In the student work, students copy and complete 12 of these patterns.
Part 2 of the first lesson is a problem-solving strategy involving writing number sentences, and thus is not part of the instruction on the division of decimals.
Lesson 2 continues by working on the division of a decimal number by a whole number. Students work with models to investigate this process. First, they divide 1.5 by three. They begin with decimal models which they cut up to show tenths. They are instructed to divide the tenths into three groups of the same size and record the results. They discuss the result, and make more models in the problem set.
Lesson 3 continues dividing decimals by whole numbers. Models are used to illustrate 3.8 divided by 2. This time, the standard algorithm is shown next to the models in four steps. Then, two examples are given using the standard algorithm only. One has some annotation. Students are instructed to discuss where to place the decimal point. The problem set contains two problems that ask for pictures, 8 that ask for the placement of the decimal point, 29 that ask for the quotient, and 24 more asking for the placement of the decimal point. A few application problems follow.
The same topic continues in lesson 4. The case of adding zeros to the dividend is addressed. Students are instructed to add zeros to divide. Three examples are given followed by the problem set.
Lesson 5 addresses choosing an operation, and is mainly concerned with selection between multiplying a decimal by a whole number and dividing a decimal by a whole number.
In summary, division of decimals only is covered through the division of a decimal by a whole number in relatively simple cases. Of the lessons in this chapter, four are directed toward this topic. Division of a decimal by a decimal is not addressed. Rounding a decimal quotient and the case of repeating decimal quotients are not covered. Explanations are modestly clear at best.
Area of Triangles [1.0]
This topic is either not present or appears to such a minimal extent as to be effectively not present.
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 [2.0]
This topic is not present in the standard portion of the student text. A treatment of exponents related to the powers of 10 appears in the student handbook portion of the text as an extension (optional) lesson. The representation of units in various place-value locations is indicated as repeated multiplications of 10. Exponent notation is introduced with arrows that point to the exponent and base. A chart is given that shows the powers of ten as numbers, as repeated multiplications, and in exponent form. Students complete some missing portions of the chart. Students answer further questions translating between repeated products of 10 and exponent notation and vice versa.
Also in the student handbook is an introduction to scientific notation as represented on calculators. Two values are indicated in standard form and scientific notation. Instructions are given in keystrokes to translate from exponent notation to standard form using two calculator brands. Four translation problems are given for solution by calculator.
Thus, the treatment of this topic only appears in a supplemental section and only addresses powers of 10 and a minimal treatment of scientific notation on calculators, not exponents and powers in general. It is likely that most students will not benefit much from this supplement.
Program Quality Evaluations
Mathematical Depth [2.8]
The mathematical depth of this program is less than desirable. Student mastery of the multiplication and division of whole numbers cannot be expected and will have to continue into the next grade level. Multiplication of decimals could also be further extend. Division of decimals is restricted to whole number divisors. The area of triangles and negative numbers are either missing or not seriously attempted. The material on powers and exponents is minimal.
Quality of Presentation [3.2]
The quality of presentation is modest. The learning objectives are often unclear, and the statements of objectives that appear are often set in context in a way that leaves the mathematics unclear and may be seen as trite or silly even by fifth grade students. The examples are generally fair, although better annotation could be provided. The explanations are sometimes unclear. On the other hand, the presentations are reasonably efficient.
Quality of Student Work [3.5]
The quantity of student work is adequate and of reasonable quality. However, the range of work is quite limited and will thus only support limited student achievement.
Overall Program Evaluation
The overall rating for this program is modest. In general, the level of mathematics supported is below expectations and student achievement will be limited. The presentation quality and student work in the program are also modest. Students may be successful to the fairly low achievement levels supported by this program.
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