In grades 9-12, the mathematics curriculum should include the continued study of algebraic concepts and methods so that all students can--

• represent situations that involve variable quantities with expressions, equations, inequalities, and matrices;
• use tables and graphs as tools to interpret expressions, equations, and inequalities;
• operate on expressions and matrices, and solve equations and inequalities;
• appreciate the power of mathematical abstraction and symbolism;

and so that, in addition, college-intending students can--

• use matrices to solve linear systems;
• demonstrate technical facility with algebraic transformations, including techniques based on the theory of equations.

Focus

Algebra is the language through which most of mathematics is communicated. It also provides a means of operating with concepts at an abstract level and then applying them, a process that often fosters generalizations and insights beyond the original context.

Aspects of this standard represent extensions of algebraic concepts developed first in grades 5-8. Whereas this earlier work was developed as a generalization of arithmetic, algebra in grades 9-12 will focus on its own logical framework and consistency. As a result, for example, algebraic symbols may represent objects rather than numbers, as in "p + q" representing the sum of two polynomials. This more sophisticated understanding of algebraic representation is a prerequisite to further formal work in virtually all mathematical subjects, including statistics, linear algebra, discrete mathematics, and calculus. Moreover, the increasing use of quantitative methods, both in the natural sciences and in such disciplines as economics, psychology, and sociology, have made algebraic processing an important tool for applying mathematics.

The proposed algebra curriculum will move away from a tight focus on manipulative facility to include a greater emphasis on conceptual understanding, on algebra as a means of representation, and on algebraic methods as a problem-solving tool. For the core program, this represents a trade-off in instructional time as well as in emphasis. For college-intending students who can expect to use their algebraic skills more often, an appropriate level of proficiency remains a goal. Even for these students, however, available and projected technology forces a rethinking of the level of skill expectations.

Discussion

Algebra as a means of representation is most readily seen in the translation of quantitative relations to equations or graphs. For example, to relate auto speed to stopping distance, collected data could be organized as in table 5.1 and analyzed for patterns.

    TABLE 5.1



    Automobile Stopping Distance



                      Reaction             Braking            Stopping



     Speed            Distance             Distance           Distance



    (in mph)          (in ft.)             (in ft.)           (in ft.)



      10                 10                    5                  15



      20                 20                   20                  40



      30                 30                   45                  75



      40                 40                   80                 120



      50                 50                  125                 175



      60                 60                  180                 240

From this particular set of data, students could deduce that if s represents speed, then the representations for the reaction, braking, and stopping distances are s, (s ²)/20, and (s ²)/20 + s, respectively. The equation d = (s ²)/20 + s would provide a problem-solving tool for interpolating and extrapolating values not included in the original table of collected data. (Corresponding activities could be applied to the equations modeling the data in each of the second and third columns of the table.) Follow-up project work for students could include preparing an oral or written report comparing this algebraic model with the usual "rule of thumb" cited in driver's education classes or researching data on braking distances for autos equipped with disc brakes or with antilock braking systems and then developing corresponding equations relating speed and stopping distance.

Situations in which there is a great amount of numerical data to be recorded and manipulated, such as with factory (store) inventories, production (sales) figures, and shipments, often are represented by matrices. For example, if I represents the initial jeans-inventory matrix (fig. 2.3), P the sales matrix, and S the shipment matrix on a given day, then I - P + S is the matrix representation of the inventory at the end of the business day. Matrix representations of data permit easy processing by computers and thus have become important representation tools in algebra.

Changes in emphases require more than simple adjustments in the amount of time to be devoted to individual topics; they also will mean changes in emphases within topics. For example, although students should spend less time simplifying radicals and manipulating rational exponents, they should devote more time to exploring examples of exponential growth and decay that can be modeled using algebra. Similarly, students should spend less time plotting curves point by point, but more time interpreting graphs, exploring the properties of graphs, and determining how these properties relate to the forms of the corresponding equations (e.g., the relationship between the graphs of y = |x| and y = |x - 5|). Of course, students should continue to plot critical points to check the reasonableness of graphs.

Computing technology enables schools to provide a richer set of algebra experiences for all students. Polynomial equations, which are very useful for describing relations among variables in a vast array of real-world situations, need no longer be a topic reserved for precalculus students. To illustrate, consider the box-building activity described in the grades 5-8 standard on communication. This activity would be extended in grades 9-12 to boxes similarly produced by cutting squares from the corners of rectangular sheets. If the dimensions of a sheet are 25 inches by 40 inches and the length of the side of the squares is x inches, then the cubic equation V = x(25 - 2x) (40 - 2x) describes the relationship between the volume and the height of the resulting box. To determine a value x for which the volume was a specified number, say, V = 1800 cubic inches, would require solving the equation x(25 - 2x) (40 - 2x) = 1800 or equivalently the equation x(25 - 2x) (40 - 2x) - 1800 = 0. Similar equations frequently arise in the management sciences in the process of analyzing cost, revenue, and profit in the production and sale of goods. Problems of this sort lead naturally to the question, "How does one solve an equation like ax ³ + bx ² + cx + d = 0?"

The following example illustrates how the treatment of polynomial equation-solving can be differentiated in both depth and the level of formalism so that all students in the core curriculum can experience success commensurate with their interests and proficiencies.

Find the roots of the equation 5x ³ - 12x ² - 16x + 8 = 0.

Level 1: Students would use either a table-building program or a graphing utility (fig. 5.1) to isolate the roots between pairs of consecutive integers.

Fig. 5.1. Bounds for roots

They would then use a successive approximation method (either a refined search by altering the input values in the table-building program or guess and check with a calculator) to estimate the roots to the nearest tenth.

Level 2: Given the conceptual understanding and information gained in level 1 activities, students would use a built-in root-finding utility by simply entering the endpoints of the appropriate unit intervals.

Level 3: Students at this level would use a graphing zoom-in process to approximate the roots to the desired degree of accuracy, subject to machine precision. Figure 5.2 illustrates how this process is used to find the negative solution with error less than 0.001. (Note that if students use only an automated zoom-in feature, this level of mathematical activity corresponds to that at level 2. The use of a zoom-in feature that requires students to interpret the scales or a viewing rectangle in selecting the appropriate x- and y-intervals for the next nested viewing rectangle requires additional mathematical sophistication. appropriate to level 3.)

Fig. 5.2. Graphing zoom-in to approximate roots

Level 4: After using a graphing utility as in level 1, students would be assigned a group project of constructing an algorithm for approximating roots, such as the following bisection algorithm. By considering several equations and investigating the relationship among equation values at, and on either side of, the midpoint of a unit interval under consideration, students can discover the pivotal idea underlying the algorithm.

To estimate, to the nearest thousandth, the root of a polynomial equation known to be in an interval ( , ) whose endpoints are consecutive integers:

(We recommend the use of notation similar to that above for expressing algorithms rather than the use of more familiar (and cumbersome) flowchart representations.)

Once an algorithm has been proposed to solve the problem, students would test the procedure by computer implementation.

Level 5: Experience with a graphing utility as in level 1 would lead to theoretical considerations regarding the number and nature of the roots. In particular, students would develop and use the rational root theorem to find the rational root(s). (In this example, 2/5 is the only rational root.) The development of the factor theorem would provide the basis for expressing this polynomial as a product of a linear and a quadratic factor, which in turn permits the other exact roots to be found by the quadratic formula. This development could be extended later to a discussion of complex roots and the fundamental theorem of algebra.

The reader should note that not only does the use of technology permit the study of polynomial equations to begin with problem situations, it also emphasizes powerful successive approximation and graphic methods that can easily be generalized to other types of equations. Moreover, the formal analysis of polynomial algebra is the culmination (level 5) of student activity, not the beginning.

Whereas all students should use matrices as tools for representation and problem solving, college-intending students also should experience formal study of matrix algebra and its applications to the solution of linear systems. Matrix methods for the solution of 2 x 2 and 3 x 3 systems are easily generalized to m equations in n variables. Computer implementation of such algorithms permits these students access to richer and more realistic problems. Further examples of the use of matrices by all students as tools for representation and problem solving are included in the elaboration for the standard on discrete mathematics.