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.
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).
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.
18.104.22.168: 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
Content Criterion 3. Mathematical terms are defined and used appropriately,
precisely, and accurately.
Content Criterion 4. Concepts and procedures are explained and
are accompanied by examples to reinforce the lessons.
Content Criterion 5. Opportunities for both mental and written
calculations are provided.
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.
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.
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.
Content Criterion 9. Selected solved examples and strategies for solving various classes of problems are provided.
Content Criterion 10. Materials must be written for individual study as well as for classroom instruction and for practice outside the classroom.
Content Criterion 11. Mathematical discussions are brought to closure. Discussion of a mathematical concept, once initiated, should be completed.
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.
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.
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.
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.
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.
Content Criterion 18. Factually accurate material is provided.
Content Criterion 19. Principles of instruction are reflective of current and confirmed research.
MEETS DOES NOT MEET
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.
Content Criterion 22. Homework assignments are provided beyond grade three (they are optional prior to grade three).
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..."
"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".