Mathematical Analysis of Top-Ranked Programs

4.1.3 Glencoe/McGraw Hill Algebra 1/Algebra 2
Relations and functions are introduced in Chapter 1, but quadratic functions are not addressed directly until Chapters 7-9. The time lag (Chapters 2-6 deal with linear equations, functions and inequalities.) might make it necessary essentially to re-teach the generic ideas at that time.

Chapter 7 deals with operations on polynomials. This is mainly a skills chapter; the word problems included seem somewhat forced. There are many exercises in each lesson (e.g., 89 exercises for lesson 7-2); it is not clear why so many similar exercises are needed. The use of algebra tiles to model operations is very nice; this sets the stage for use of this representation in Chapter 8 for factoring of trinomials . This model is explicitly tied to both horizontal and vertical symbolic recording processes for the operations on polynomials. One concern here is that students will not have much motivation to learn the skills, so they may try to memorize (rather than learn) the skills. The sequencing of the lessons and the presentation of the mathematics would seem to encourage this approach. Providing a rationale for learning this material would be a welcome addition.

Chapter 8 deals with factoring and solving quadratic equations. Again, this material is approached mainly as a sequence of skills, rather than with some underlying conceptual underpinning. Ideas addressed include factoring monomials (Lesson 8.1), factoring using the distributive property (Lesson 8.2), and factoring trinomials (Lesson 8.3). It is important that general trinomials (i.e., ax^2 + bx + c) are addressed first, initially through the model provided by algebra tiles. Differences of square and perfect squares are presented as special cases of the general case. This seems to be a good approach, since it puts the emphasis correctly on general ideas.

Chapter 9 deals with quadratic and exponential functions, though more emphasis is given
to quadratic functions here. Lesson 9-1 introduces graphs of quadratic functions and simply states “facts” about quadratic functions (e.g., the axis of symmetry is x = -(b/2a)), without providing a clear rationale for why these facts are true. This approach would seem to encourage students to memorize information rather than trying to understand that information. Subsequent topics include solving by graphing (Lesson 9-2), transformations (lesson 9-3), completing the square (Lesson 9-4), and quadratic formula (Lesson 9-5). Lessons 9-6 through 9-9 provide experience with exponential functions and finite differences. As in earlier chapters, there are many exercises (e.g., 95 for Lesson 9-1), without any obvious reason for so many.

The sequencing of ideas in this Algebra 1 book is quite traditional. There seems to be an over-emphasis on skill development rather than conceptual development. However, this approach lends itself to a relatively close alignment of the book to almost any set of standards. The sequence of lessons would be understandable to most high school mathematics teachers, even though it might not generate a coherent “view” of mathematics ideas among novices (i.e., students).

Algebra 2 addresses quadratic functions mainly in Chapter 5. The work from Algebra 1 is revisited, with extensions of some work to complex numbers . In this course, too, some key facts (e.g., “A quadratic equation can have one, two, or no real solutions.” p. 260) are simply stated, without any rationale, other than examples, for why those facts are true. If teachers do not emphasize the examples adequately, this approach would seem to encourage memorization. The development of transformations of quadratic functions is done more completely here than in the earlier book.

Chapter 6 addresses operations (including division) on polynomials, and polynomial functions. This work goes beyond that required by the Algebra 2 Standards, but it is organized to help students gain insight into an important set of mathematical ideas (e.g, rational zero theorem). This seems to be a nice extension of work with quadratic functions. Lesson 10-2 also deals with parabolas as part of the study of conic sections .

Overall, the mathematics is sound, though there is probably not enough rationale provided for helping students want to learn the mathematics. The approach is heavily oriented toward skill development.

4.1.4 Prentice Hall Algebra1/Algebra 2
In Algebra 1 the concept of function is introduced in chapter 1, along with domain and range. This lays general background for later work, even though there is not much development here.

Functions reappear in much more depth in Chapter 5, which is a general discussion of functions. First, functions are used as models for events (Lesson 5-1). This is followed by relations and functions (Lesson 5-2), rules, tables, and graphs (Lesson 5-3), and four lessons on writing and using function rules . These four lessons seem to present the mathematics as compartmentalized ideas , somewhat disjoint from each other. There is no apparent underlying common thread that ties the ideas together.

Chapter 9 is focused on operations on polynomials and factoring. Algebra tiles are used as a model for multiplication of binomials , with connections made to both vertical and horizontal recording schemes. Factoring is introduced first for x^2 +bx + c (i.e., finding factors of C whose sum is b; Lesson 9-5) and then ax^2 +bx + c (i.e., “reverse application of FOIL”; Lesson 9-6). Special cases of difference of two squares and perfect squares (Lesson 9-7) are presented through rules as well as
examples. Algebra tiles are used in an activity lab, but do not appear as part of the primary focus on instruction.

Chapter 10 begins with graphing of special cases of quadratic functions (Lessons 10.1), namely, y = ax^2 and y = ax^2 + c. Then the general case is presented (Lessons 10.2), along with graphing of inequalities. It is not clear why the special cases need to be presented first. There is a short demonstration that attempts to justify the equation of the axis symmetry. In Lesson 10-3 quadratic equations are solved by graphing, along with use of square roots to solve ax^2 + c, but these strategies are not connected in any way. Lesson 10-4 is factoring to solve quadratic equations, followed by completing the square (Lesson 10-5), quadratic formula (Lesson 10-6), discriminant (Lesson 10-7), and modeling (Lesson 10-8). Instruction is through worked-out examples followed by exercises. The mathematics is correct, and the sequence would probably be comfortable to most high school mathematics teachers, but there is very little help provided for students in understanding how these ideas and skills tie together. Ideas are presented in a compartmentalized way.

In Algebra 2, the work is reviewed and extended. There is still a tendency to reduce ideas to a series of “cases.” For example, Lesson 5-4 on factoring has worked-out examples for several cases: (1) ac > 0 and b > 0, (2) ac > 0 and b < 0, (3) ac < 0, (4) a ≠ 1 and ac > 0, and (5) a ≠ 1 and ac < 0. This could clearly create the impression that identifying what case “applies” is the first step in determining how to factor a trinomial, followed by applying some memorized procedures for that case. This makes the issue of factoring an overwhelming learning burden. The major extension in this chapter is work with complex numbers, so that completing the square and quadratic formula work can include
imaginary solutions .

Overall, the mathematics is sound, though there is not enough rationale provided for helping students want to learn the mathematics. The sequencing of examples and procedures tends to create an impression that there are many distinct “cases” that students should remember. There is too little attempt to “combine” cases under some general umbrella so that students understand how the cases are related to each other.

4.1.5 Conclusions: Algebra 1/Algebra 2
All four series provide coverage of mathematically sound content. The Discovering series and the Holt series seem to be the ones that tie together key mathematics ideas best. Since coherence of mathematics ideas is a part of mathematical soundness, these two series rate high. The Glencoe and Prentice Hall series leave an impression of compartmentalization of ideas. These two series rate somewhat lower, though they are still mathematically sound. Teachers might have to work harder to ensure that students develop deep understanding.

4.2 Geometry

One of the major themes in the Geometry standards is proof. It is clearly important to develop the idea of proof rigorously. One other major theme in Geometry is continued development of properties of figures . We have chosen to focus on parallel/perpendicular lines and parallelograms. The relevant Performance Expectations are listed below.

G.1.A (M1.4.A)
Distinguish between inductive and deductive reasoning.

G.1.B (M1.4.B)
Use inductive reasoning to make conjectures, to test the plausibility of a
geometric statement, and to help find a counterexample.

G.1.C (M1.4.C and M2.3.A)
Use deductive reasoning to prove that a valid geometric
statement is true.

G.1.D (M2.3.C)
Write the converse, inverse, and contrapositive of a valid proposition
and determine their validity.

G.1.E (M2.3.B)
Identify errors or gaps in a mathematical argument and develop
counterexamples to refute invalid statements about geometric relationships.

G.1.F (M2.3.D)
Distinguish between definitions and undefined geometric terms and
explain the role of definitions, undefined terms, postulates (axioms), and
theorems.

G.2.A (M1.4.E)
Know, prove, and apply theorems about parallel and perpendicular lines.

G.2.B (M1.4.F)
Know, prove, and apply theorems about angles, including angles that
arise from parallel lines intersected by a transversal.

G.2.C (M1.4.G)
Explain and perform basic compass and straightedge constructions
related to parallel and perpendicular lines.

G.3.F (M2.3.J)
Know, prove, and apply basic theorems about parallelograms.

G.3.G (M2.3.K)
Know, prove, and apply theorems about properties of quadrilaterals and
other polygons.

G.4.A (M1.3.H)
Determine the equation of a line in the coordinate plane that is described
geometrically, including a line through two given points, a line through a given
point parallel to a given line, and a line through a given point perpendicular to a
given line.

G.4.B (M2.3.L)
Determine the coordinates of a point that is described geometrically.

G.4.C (M2.3.M)
Verify and apply properties of triangles and quadrilaterals in the
coordinate plane.

What is called for is a set of theorems stating properties of parallelograms. What is needed for this are the basic theorems about angles formed by parallels and a transversal, along with the angle sum theorem for polygons and some congruence theorems for triangles. In the reviews that follow, these topics will be referred to as the standard parallelogram theorems.