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
This program does not incorporate a student text of the standard variety. Instead, the student encounters the program largely through consumable materials. There are two soft-cover Journals that contain this material for grade 5. The journals contain materials for student work and a small glossary, but the extent of exposition is often minimal. Journal work is typically done in class. Masters for worksheets called Study Links are also provided. The study links are typically done as homework. The Teacher's Manual & Lesson Guide (two volumes) directs the implementation of the program, although teachers will also have to study the Teacher's Reference Manual which contains teacher explanations for many of the concepts and procedures used in grades 4-6.
The Teacher's Manual reveals that the grade 5 program is divided into 11 units:
Each unit contains from 8 to 15 lessons. There are a total of 125 lessons in the program, which includes an assessment at the end of each unit.
Lessons are often a mixture of whole class, small group, and individual work. For example, lesson 35 in unit 4 is called Two Rules for finding equivalent fractions. The lesson begins with a brief review of the prior worksheet. The first activity is based on a journal page that the students work on in pairs and then discuss the answers. In a whole class discussion in which the teacher is to Help students to formulate the multiplication rule, which is that they can multiply the numerator and denominator by the same number to get an equivalent fraction. A division rule is also developed in a similar manner. Students then work a follow-up journal page in pairs. Some students may then work on additional activities while the teacher helps those with difficulties. The study link (which would represent homework in most cases) consists of:
Content Area Evaluations
Multiplication and Division of Whole Numbers [2.5]
Multiplication of Whole Numbers
In this text, multiplication is represented by * and division is represented by /.
The bulk of the material addressing whole number multiplication appears in the first three units of the program, primarily in Unit 2.
In Unit 1, students model multiplication as arrays, review the multiplication facts and study the relationship between multiplication and division. Students are also introduced to factors of numbers.
In Unit 2, lesson 11 introduces estimation that may involve multiplication and begins rehearsal of multiplying by powers of 10 to extend the basic facts, although the technique is without explanation.
Two lessons are devoted to multiplication algorithms. In Lesson 16, using a journal page, teachers review two algorithms. One is a partial-products technique that is preferred by this program. At this stage, the two factors are re-written in expanded form. Partial products are written beneath as in the standard algorithm (training zeros are included). However, a partial product is written for each pair of digits that are multiplied. This process eliminates the regrouping and instead the regrouped (carried) sums are computed while adding up the partial products. On the other hand, the addition of partial products can become a large problem. For example, there would be a column of 12 number to add up in a 3-digit by 4-digit multiplication problem.
The second algorithm introduced is the lattice method. Students then work in pairs to solve 6 multiplication problems from 2-digits by 2-digits up to 2-digits by 3-digits and 1-digit by 4-digits. Then, they practice estimating products in a game format.
Lesson 17 introduces an ancient Egyptian multiplication algorithm. In this method, successive doublings of the second factor are listed in one column and binary powers are listed in the other. Then, the binary numbers that compose the first factor are identified and the rest of the rows are crossed off. The sum of the remaining doublings is then computed to yield the product. Students explore this algorithm and then are to list the advantages and disadvantages of the three algorithms they have covered.
The assessment for Unit 2 includes 6 numeric multiplication problems from 1-digit by 2-digits up to 3-digits by 2-digits. It also includes 2 story multiplication problems, and students also make up and solve a problem that might use multiplication.
In Unit 3, students are shown a technique for multiplications when one factor ends in 9 (rounding up and subtracting later), and 7 problems are worked. A few problems including multiplying by powers of 10 are included.
A small multiplication drill or game appears about once per unit in the following units of the program.
Thus, the coverage in this program is limited to multiplication by 2-digit factors. Even this is given rather minimal support reflecting a decreased attention to algorithms and practice in the program.
Division of Whole Numbers
In the first part of the program, a few exercises ask students simple division problems, like how many 40s are in 160?
Work on division largely appears in Unit 6. In the first lesson of this unit, students review division facts and their multiple-of-10 extensions. Seven story problems are then solved. Dividing the problem into two simpler divisions and summing the result is then introduced as a technique to extend mental division through a whole-class discussion. Students then play a game involving division of two-digit numbers by a 1-digit number.
The next lesson introduces the division algorithm preferred by this program. It is similar to the standard algorithm, except that the partial quotients are written in a column to the right of the problem. In this way, they are written in their full (expanded) form, that is the trailing zeros are included. Note that in using this method, the partial quotients need not be multiples of powers of 10. The algorithm is introduced in whole class discussion, and the terms quotient and remainder and divisor are introduced. Students work in pairs to solve three division problems with 1-digit divisors. In the third lesson of this unit, journal pages include 10 division problems. The next lesson discusses remainders.
The assessment for this unit includes 6 numeric division problems and 5 division story problems.
A few small exercise sections appear in the remaining units, mostly practice with 1-digit divisors.
In summary, this program begins work with multiplication on simple arrays and multiplication facts. The standard algorithm is not introduced, but a partial-products algorithm that has a regular recording process is used instead. No special attention is given to growing the multiplication skill to larger and larger problems. At about mid-year, students are introduced to a division algorithm that is similar to the standard algorithm and are given a lesson on remainders. No special attention is given to working larger and larger problems. The presentation of multiplication and division feels casual in that it is not given a lot of attention or a lot of lessons.
Decimal Multiplication and Division [1.1]
Decimal Multiplication
Decimal multiplication is not directly covered. A small portion of this topic does appear as multiples of 0.1 are used in the explanation of negative exponents. However, this topic appears to such a minimal extent as to be effectively not present.
Decimal Division
This topic is either not present or appears to such a minimal extent as to be effectively not present.
Area of Triangles [2.7]
The area of triangles appears at about mid-year. Lesson 68 introduces the rectangle method for finding the area of triangles and parallelograms. Students work with partners to find areas of two triangles and a parallelogram with no explanation.
Working as a whole class, the teacher explains two methods using rectangles. First, working with an acute triangle, the triangle is divided into two right triangles so that a rectangle can be drawn for each half. The area of the inscribed triangle is half the area of the rectangle in each case, so the area of the original full triangle is half the sum of the areas of the two rectangles. Students will likely see that this is the case from the journal page.
The second method is illustrated with an obtuse triangle. A rectangle is drawn around the entire triangle. This forms two areas within the rectangle that are not part of the triangle. One of them will be half of the rectangle, so the area can be computed as half that of the rectangle. The second area will be half that of a smaller rectangle. The area of the object triangle is thus the area of the rectangle minus the two computed right-triangle areas. Students will be less likely to see that this is the case from the journal page. This subtract the excess method is then extended to parallelograms. A journal page with practice problems then contains three triangle and three parallelogram problems.
In the next lesson, students discuss their definitions of base and height as a whole class activity based on what they wrote on a journal page, but they are not given a written definition. Students work with partners to identify bases and heights for triangles and parallelograms, and then to complete a table of lengths and areas. Students in pairs are then asked to generate the formulas for area for triangles and parallelograms. These formulas are then discussed as a whole class activity. A homework sheet asks for the area of a complex polygon, likely to be computed as the sum of the areas of four parallelograms.
The subsequent lesson introduces Pick's formula for approximating the area of polygons on grids.
Negative Numbers [3.2]
The topic of negative numbers first appears about mid-year in Unit 6. The topic is introduced with a whole class discussion in which students to share their characterizations of people with positive and negative attitudes, presumably to show opposites.
The whole class continues by reading a journal page that introduces negative numbers and their notation. A table is give with six situations in which negative and positive numbers are used:
temperature, business (loss and profit) bank accounts (withdrawal and deposit), time (before and after), games (behind and ahead), elevation, and gauges, dials, and dipsticks. It is noted that negative numbers and positive numbers are opposites, and these are shown on the number line. The teacher may choose to mention the +/- calculator key at this point.
Students then work a journal page with partners, identifying positive and negative points on the number line and writing a few positive and negative numbers. On a second journal page completed in pairs they write more numbers and answer simple questions about them. The study link assignment contains a few more simple questions about negative numbers.
In Unit 8, there is a lesson on the addition of positive and negative numbers and another lesson on subtraction. These lessons work with partners or small groups to solve these problems using + and - counters (e.g., black and red squares). They then come together for whole class discussions for the development of the rules of addition and subtraction for these cases. For example, the Teacher's Manual and Lesson Guide states If both addends have the same sign (both + or both -), add the number parts and write the sign of the addends in front of the sum and if the addends have unlike signs (a + and a -), subtract the smaller from the larger number part and write the sign of the addend with the larger number part in front of the answer. Students work a few more problems without counters in a study link.
The next lesson introduces the use of a slide rule to add and subtract. These are integer subtraction slider and integer addition slider and integer holder strips of paper which accomplish number-line-like additions and subtractions. The next lesson then introduces the use of the calculator operations needed to do addition and subtraction with negative numbers. A few lessons later, students identify some points on the 4-quadrant coordinate grid, but this material is given only minor attention. Consequently, these three lessons contribute little to the understanding of or computations with negative numbers.
Thus, negative numbers are important in six lessons. One lesson essentially on the meaning of negative numbers, one on addition, and one on subtraction constitute the introduction to this material. The slide rule lesson and the calculator lesson don't add much conceptually to the topic coverage. The extension of the coordinate grid to four quadrants receives minor attention.
Powers, Exponents and Scientific Notation [4.2]
In Unit 1, exponential notation for squared numbers is introduced. A journal page explains the notation. The terms exponent and exponential notation are defined, but base is not defined. Although the text of the journal page does not imply that greater exponents cannot be used, this page is restricted to squared terms. Students work a small number of problems translating both ways.
Students then learn about the x2 key on the calculator (if their calculator has one), and use the calculator to find some squares.
Further work in this unit covers using the exponent notation to represent factorization of numbers. A study link (probably homework) page again explains the notation conventions. Again reference is made to exponents and exponential notation but not to bases. This page is addressing prime factorization. Students then translate from exponential notation to standard notation in 4 problems, and write prime factorization in exponential notation in 4 more problems.
Some mention of exponents also appears in the discussion of powers of 2 that underlie the ancient Egyptian algorithm for multiplication in Unit 2, but the discussion of powers is really tangential.
Exponential notation is covered in greater detail in Unit 6. The first journal page explains the notation system. The terms exponent, exponential notation, and base are explained. Students complete a table that shows exponential notation, the base, the exponent, the repeated multiplication, and the resulting product.
In the next journal page, students practice evaluating exponents using a calculator to do repeated multiplications.
On the same page they are introduced to the yx calculator key. Students are given five problems which consist of sequences of keystrokes. They are to write down the resulting display. Then, they are asked what the function of the yx key is. There is no explanation offered.
The lesson ends with a game to practice with exponents. The study link page explains exponential notation briefly. Four problems are given in which the translation between exponential notation and standard notation has been done incorrectly. Students are to find and correct the error.
Later in the unit, they are introduced to exponential notation for powers of 10. In a whole class activity, they read a journal page that discusses powers of 10 and the common prefixes used to represent them. They answer 6 questions about this material. Students work with partners to answer four more questions translating between exponential notation for powers of 10 and their standard form written as words.
In the same lesson, the class reads part of a journal page that introduces Powers of 0.1 or negative powers of 10. Examples show that 10 raised to a negative exponent is 0.1 raised to the positive of that power. A table shows 4 values written in words, exponential notation, and as a prefix. A few questions about this representation are given. This is not a meaningful introduction to negative exponents.
In the next lesson, students read an introduction to scientific notation as a whole class. They work with partners to translate between scientific notation and standard notation and vice versa.
The next journal pages cover translating from scientific notation to standard notation in a problem set done with partners.
A whole class activity then explains how to read the calculator representation of scientific notation. The journal page explains this process briefly, and students then complete a table of information.
A game using scientific notation is played.
Later, in unit 7, the value 4 is raised to the 0 power in the context of a game, but this is not a good treatment of this topic.
In summary, a lot of student time is devoted to this topic. A positive feature is that students are to find and correct errors in translating between notation systems. On the other hand, coverage of advanced topics such as a power of zero and negative powers, is minimal and not sufficient for student understanding.
Program Quality Evaluations
Mathematical Depth [2.3]
The mathematical depth of this program is variable across topic areas in a curious way. The multiplication and division of whole numbers, which should be reaching mastery levels, actually is given rather minimal coverage. Contents really only address the 2-digit case and then in a rather casual way that is unlikely to support mastery. Decimal multiplication and division are virtually absent from the program. When it comes to the area of triangles, on the other hand, students are expected to generate the formula themselves. Negative numbers are introduced up to a minimal treatment of addition and subtraction operations. Material with respect to powers and exponents seems curiously misaligned with other topics as it introduces negative powers of 10 as repeated multiplications of 0.1. This seems odd for a program that defers treatment of the multiplication of decimals until the next year. Thus, despite some surprising treatment of the more advanced topics in the review, the weak treatment overall results in a low rating for the mathematical depth of the program.
Quality of Presentation [2.7]
Along with the low rating of Mathematical Depth and the lack of a standard student text, this program was seen as having a rather weak presentation quality. Perhaps the strength of the program is in the quality of the examples dictated in the teacher's materials and in the student worksheets. Sometimes the student sheets provide instruction or exposition as well as sample problems, but this is not frequent enough. The objectives of instruction will not always be clear to students as they go through the process. The efficiency of instruction is somewhat low, and the support for student mastery is weak in this program.
Quality of Student Work [2.2]
Despite the number of student sheets provided, the student work expected in this program is limited both in sheer quantity and also in range and scope. In several content areas, the level of student work appears quite limited suggesting that effective learning will be at risk.
Overall Program Evaluation
The overall evaluation of this program was next-to-lowest among the fifth grade programs in this review. The program comes across with the flavor of a survey of some rather sophisticated areas of mathematics for fifth-grade students without support for the development of topics or student mastery of content. This unusual combination of features makes it difficult to imagine a fifth-grade circumstance where such a program could be recommended.
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