PRELIMINARY CONTENT REVIEW PANEL REPORT
Publisher: Prentice Hall Title: Algebra I
Reviewer’s Name: Wayne Bishop, Jane Friedman, Steve Kerkhoff, Yat-Sun Poon
 

Conclusion
 

A succinct statement summarizing the CRP’s evaluation.


With regard to mathematics content only, this program sufficiently addresses the content standards and applicable evaluation criteria to be recommended for adoption as submitted.

This is a solid text that sufficiently addresses all the Standards. It
provides clear explanations throughout and its few more formal proofs are
done carefully and clearly in a manner that allows students to be
comfortable with this style of argument. It has many routine and
skill-building problems at the end of each section as well Mathematical
Reasoning and Application problems. The text is weakened by an unnecessary
restriction to rational numbers in many of the earlier chapters. Although
real numbers are introduced and discussed sufficiently to provide weak
coverage of this Standard, it would be quite easy to change the earlier
chapters to include general real numbers. This would greatly improve the
text; it should be done in the next edition.

Each section contains several worked examples with step-by-step explanations
and the attention paid to Mathematical Reasoning is quite adequate. Clear
pacing instructions are provided for teachers for both traditional and
transitional students.  This is a very usable text.


 

Mathematics Content/Alignment with Standards

A systematic review of determinations regarding the criteria in this section.  Citations of standards not adequately addressed (if any) are of particular importance with regard to Content Criterion 1.


Content Criterion 1.  The content supports teaching the mathematics standards at each grade level (as detailed, discussed, and prioritized in Chapters 2 and 3 of the framework).
MEETS


A standard by standard summary follows:
1.0: Good discussion of basic properties and axioms, including the closure
condition for various subsets. Unfortunately, the longest and most detailed
section covering this topic unnecessarily restricts itself to rational
numbers, rather than all real numbers. The properties are extended to real
numbers only in a brief section in Chapter 11. The discussion of the
difference between irrational and rational numbers is adequate, even going
so far as to prove that the square root of 7 is irrational. However,
restriction to rational numbers for much of the text (though, of course, one
expects virtually all examples to involve only rationals) is probably the
text's weakest feature.

2.0: Adequate, though short, discussion of fractional exponents.  Again, the fact that the exponent laws (for integer exponents) are first stated only for rational base, makes the later discussion of fractional exponents awkward (e.g., does the square of the square root of 2 make sense).  Strictly speaking, this issue is dealt with, but not particularly well.

3.4.5.6: These important standards are covered quite well, being developed over many pages, with plenty of practice problems and with examples worked out explicitly in the text.

7,8: Short, but to the point.

9: Covered thoroughly and well with a good blending of geometric and algebraic methods of solving the equations.

10: Adequate, but could use more word problems.

11: The factoring rules are presented clearly with plenty of examples and practice problems. Also, rules are given in abstract algebraic form. The one weakness is that all factoring implicitly seems to be taking place over the integers rather than allowing fractional coefficients.  The fact that the algebraic formulae are there makes up for this to a certain extent, but this issue could lead to possible confusion. (See Corrections and Edits at the end of this report.)

12, 13: Sufficient coverage.

14, 19, 20, 21, 22, 23: The material leading up to and including the explanation of the quadratic formula by completing the square is well-done, with a gradual increase in the complexity of the examples. This final proof is illustrated thoroughly by explicit examples. Relation to root finding, the geometry of the graphs, and the role of the discriminant are all sufficiently covered. There should be more applications/word problems for quadratic functions.

16, 17, 18: Adequate coverage of relations and functions though the discussions of the domain and range, particularly of the square root function are weak.  However, coverage of some of the other specific types of functional relationships (e.g., direct and indirect variation) is pretty good, with plenty of examples and word problems.

15: Coverage of rate and ratio problems is sufficient. However, the percent mixture problems are awkwardly place between unrelated topics.

Content Criterion 2.  A checklist of evidence accompanies the submission and includes page numbers or other references and demonstrates alignment with the mathematics content standards and, to the extent possible, the framework.
MEETS


The Standards map for this text is generally accurate and demonstrates alignment with the Standards.

Content Criterion 3.  Mathematical terms are defined and used appropriately, precisely, and accurately.
MEETS


Mathematical terms are generally defined and used accurately in this text. However, on page 35, the definition of "Equivalent Equations" is not quite correct.  The definition should be in terms of the four operations that are listed there applied to both sides of the equation.  Having the same solution set is a consequence of this definition; it should not be the definition itself.  Otherwise any two equations that, for example, have no solutions would be "equivalent". (The notions turn out to be the same for linear equations, but not in general.)
The definition can be fixed by a straightforward correction and the way in which the term is used elsewhere in the text is, in fact, correct and precisely in terms of the definition given  in the Corrections and Edits section at the end.

Content Criterion 4.  Concepts and procedures are explained and are accompanied by examples to reinforce the lessons.
MEETS


Virtually every new concept, idea, and problem type that is introduced is immediately followed by a well-chosen example.  In these examples, each step is carefully explained. This is one of the strengths of this text.

Content Criterion 5.  Opportunities for both mental and written calculations are provided.
MEETS


Every section has many written calculation problems. A few have mental calculation problems; these are less important at this grade level.

Content Criterion 6.  Many types of problems are provided: those that help develop a concept, those that provide practice in learning a skill, those that apply previously learned concepts and skills to new situations, those that are mathematically interesting and challenging, and those that require proofs.

MEETS


Several types of problems ("Mathematical Reasoning", Critical Thought", "Error Analysis") are included in most sections, and, for the most part, their titles are an accurate reflection of what they are.

Content Criterion 7.  Ample practice is provided with both routine calculations and more involved multi-step procedures in order to foster the automatic use of these procedures and to foster the development of mathematical understanding, which is described in Chapters 1 and 4.

MEETS


Yes, See Criterion 6.

Content Criterion 8.  Applications of mathematics are given when appropriate, both within mathematics and to problems arising from daily life. Applications must not dictate the scope and sequence of the mathematics program and the use of brand names and logos should be avoided. When the mathematics is understood, one can teach students how to apply it.

MEETS


There are plenty of examples coming from daily life. The sections (in green) called "Application" are of particular interest.

Content Criterion 9.  Selected solved examples and strategies for solving various classes of problems are provided.

MEETS


Yes. See Content Criterion 4.

Content Criterion 10.  Materials must be written for individual study as well as for classroom instruction and for practice outside the classroom.

MEETS


The pacing guide, math background, teaching notes, and assignment guide in the Teacher's Edition support classroom instruction, but could also be used for individual study outside the classroom. The pacing guide offers teachers and individuals the option of moving through the same material at a slower pace for transitional students. The student text itself is quite readable and makes it quite clear what is to be learned in each section so that the student is not completely dependent on a teacher or parent for explanations.  Hence, it could be used successfully outside a classroom.

Content Criterion 11.  Mathematical discussions are brought to closure. Discussion of a mathematical concept, once initiated, should be completed.

MEETS


The main points of each section of a chapter are generally made quite explicit. There is the very nice feature of the "Chapter Wrap-Up" that summarizes each section in the chapter.

Content Criterion 12.  All formulas and theorems appropriate for the grade level should be proved, and reasons should be given when an important proof is not proved.
MEETS


This text does a very good job of explaining things whenever it can and making it clear when it is using something or stating something beyond what it can show at the time. For example, it gives a proof that the square root of 7 is irrational (page 486) which is very clear. Along the way, it uses without proof, that a prime factor of the square of a number is a prime factor of the number itself. But it is quite clear about exactly what fact it is using and that it is not going to try to justify it.

Another good example is the excellent proof that "m" in the linear equation "y = mx+b" is the slope in the sense of the "rise over the fun." This is an issue that is often avoided unnecessarily in texts at this level; the explanation in this text is a model of clarity.

Content Criterion 13.  Topics cover broad levels of difficulty. Materials must address mathematical content from the standards well beyond a minimal level of competence.

MEETS


Most of the standards are covered thoroughly at a level well beyond competence.

Content Criterion 14.  Attention and emphasis differ across the standards in accordance with (1) the emphasis given to standards in Chap--ter 3; and (2) the inherent complexity and difficulty of a given standard.
 MEETS     DOES NOT MEET


 

Content Criterion 15.  Optional activities, advanced problems, discretionary activities, enrichment activities, and supplemental activities or examples are clearly identified and are easily accessible to teachers and students alike.
MEETS


Many of the sections have problems labeled "Challenge" that extend beyond the basic content of the text.  There are also often "Connections" and "Applications" sections that relate the material of the previous sections to other topics, generalize them beyond the immediate grade-level requirements, or ask open-ended questions.

Content Criterion 16.  A substantial majority of the material relates directly to the mathematics standards for each grade level, although standards from earlier grades may be reinforced. The foundation for the mastery of later standards should be built at each grade level.

MEETS


Yes. The text does not spend too much time on review material. However, each chapter has a sections called "Skills and Concepts You Need" that allows students and teachers to do a quick check for any review that might be necessary.

Content Criterion 17.  An overwhelming majority of the submission is devoted directly to mathematics. Extraneous topics that are not tied to meeting or exceeding the standards, or to the goals of the framework, are kept to a minimum; and extraneous material is not in conflict with the standards. Any non-mathe-matical content must be clearly relevant to mathematics. Mathematical content can include applications, worked problems, problem sets, and line drawings that represent and clarify the process of abstraction.

MEETS


There is not a lot of extraneous material in the text. However, the pages are too busy, including the parts dealing with mathematics. The removal of distracting visual material, including unnecessary pictures and overuse of color, would improve the text and allow the students to concentrate on the mathematics. Practice problems for multiple choice tests put the focus on test-taking rather on mathematical content. they would be better off in the Teachers' Edition only.

Content Criterion 18.  Factually accurate material is provided.

MEETS


There do not seem to be many errors in the text.

Content Criterion 19.  Principles of instruction are reflective of current and confirmed research.

MEETS    DOES NOT MEET


The CRP members generally agreed that they would not comment on this criterion.

Content Criterion 20.  Materials drawn from other subject-matter areas are scholarly and accurate in relation to that other subject-matter area. For example, if a mathematics program includes an example related to science, the scientific references must be scholarly and accurate.
 MEETS    DOES NOT MEET


 

Content Criterion 21.  Regular opportunities are provided for students to demonstrate mathematical reasoning. Such demonstrations may take a variety of forms, but they should always focus on logical reasoning, such as showing steps in calculations or giving oral and written explanations of how to solve a particular problem.

MEETS


There are Mathematical Reasoning problems, usually accurately labeled, in each section. The examples given in the text generally have clear  explanations for each step in a derivation.

Content Criterion 22.  Homework assignments are provided beyond grade three (they are optional prior to grade three).

MEETS


There are homework assignments, using problems in the text, suggested in the Teachers' Edition for every section.

Additional Comments and Citations.
 
 

Corrections and Edits.

A. As discussed in Content Criterion 3, the definition of equivalent equations (page 35) should be in terms of the four operations listed there. Having the same solution set is a consequence of this definition, rather than the definition itself. This can be easily remedied by the following corrections and rearrangement of sentences:

After the sub-heading: "Objective: Recognize equivalent equations." begin the definition with:

"Two equations are equivalent if one can be obtained from the other by a sequence of the following steps:"

Then put in the paragraph reading:

"You can add the same number to both..."

down to
   "divide both sides...same nonzero number."

In a separate paragraph state:

"Equivalent equations have the same solution set."

Then put in the remainder of the section beginning with the double headings:

Equivalent equations              Non-equivalent equations

On page 49, in the "Wrap-Up" section this change of definition needs to be reflected by changing the last sentence of section 1-7 to:
"Two equations are equivalent if one can be obtained from the other by a sequence of the steps listed on page 35. Equivalent equations have the same solution set."

B. In the Teachers' Edition, on the left hand side of page 270, there are confusing explanations for why two different trinomials are not perfect squares. Change the explanation for x^2 +5xy + 1 to read:

"...is not a perfect square because the middle term would have to be +2x or -2x".
The explanation for x^2 + 3xy +y^2 (#2) should read
"...is not a perfect square because the middle term would have to be +2xy or -2xy".