In grades 9-12, the mathematics curriculum should include investigation of the connections and interplay among various mathematical topics and their applications so that all students can--

• recognize equivalent representations of the same concept;
• relate procedures in one representation to procedures in an equivalent representation;
• use and value the connections among mathematical topics;
• use and value the connections between mathematics and other disciplines.

Focus

This standard emphasizes the importance of the connections among mathematical topics and those between mathematics and other disciplines, connections that are alluded to in many of the other standards. Two general types of connections are important: (1) modeling connections between problem situations that may arise in the real world or in disciplines other than mathematics and their mathematical representation(s); and (2) mathematical connections between two equivalent representations and between corresponding processes in each. These connections are illustrated in figure 4.1.

Fig. 4.1. Two general types of connections

Students who are able to apply and translate among different representations of the same problem situation or of the same mathematical concept will have at once a powerful, flexible set of tools for solving problems and a deeper appreciation of the consistency and beauty of mathematics.

It is important to recognize that the effectiveness of informal exploration depends on the extent to which it enables students to recognize crucial connections between informal activities and the mathematical ideas those activities are meant to convey. For example, exploratory geometric constructions help to develop understanding of formal geometric concepts only if students grasp the connections between straightedge and compass procedures and their formal geometric analogs.

Discussion

As in the earlier grades, teachers in grades 9-12 should introduce a new topic by exploring appropriate concrete representations in which students recognize that the exploratory activities embody the mathematical topic. This helps establish modeling connections, which can be further strengthened by an instructional approach that encourages multiple methods of solution for any given problem. Students should constantly be encouraged to look back at the solution process to consider other possible strategies. Such an approach helps establish mathematical connections as well by focusing students' attention on commonalities across different mathematical representations.

Students' understanding of the connections among mathematical ideas facilitates their ability to formulate and deductively verify conjectures across topics, an activity that becomes increasingly important in grades 9-12. In turn, these newly developed mathematical concepts and procedures can be applied to solve other problems arising from within mathematics and from other disciplines. The pervasiveness of the connections between mathematics and other disciplines is only hinted at by the following brief list of applications:

• Art: the use of symmetry, perspective, spatial representations, and patterns (including fractals) to create original artistic works
• Biology: the use of scaling to identify limiting factors on the growth of various organisms
• Business: the optimization of a communication network
• Industrial arts: the use of mathematics-based computer-aided design in producing scale drawings or models of three-dimensional objects such as houses
• Medicine: modeling an inoculation plan to eliminate an infectious disease
• Physics: the use of vectors to address problems involving forces
• Social science: the use of statistical techniques in predicting and analyzing election results

As a means of emphasizing the connections between mathematical ideas, new concepts should be introduced, where possible, as extensions of familiar mathematics. For example, the definitions of the trigonometric functions in terms of the coordinates of a point on the terminal side of an angle in standard position should be viewed as a natural extension of the trigonometric ratios defined for right triangles. In addition, the relationships among different representations of the same concept should be explored. Students, for example, may discover that the motion of a half-turn in geometry (fig. 4.2(a)) can be represented as a composition of reflections across two perpendicular lines or as the function H[(x, y)] = (-x, -y) (fig. 4.2(b)), and as a matrix product (fig. 4.2(c)).

Fig. 4.2. Multiple representations of a half-turn

The connections between algebra and geometry are among the most important in high school mathematics. For example, finding | a - b | corresponds to determining the distance between coordinates a and b on a number line; finding a zero of a function in algebra corresponds to determining an x-intercept of the graph of the function; finding a solution to a system of equations in algebra corresponds to determining the coordinates of the point(s) of intersection of the graphs of the equations; determining a local maximum value of a function in algebra corresponds to determining a local high point on the graph of the function; expressing a function in the form y = af(bx + c) + d identifies the transformations that relate its graph to that of the simpler function y = f(x); and approximating the limit of a function at a point of discontinuity corresponds to investigating the behavior of the graph of the function near that point. Computer-graphing technology now makes it possible to exploit these connections; many problems traditionally solved using algebra can now be solved efficiently and in more general cases using the geometric representation and computer-graphing techniques. This approach removes algebraic manipulative skill as a prerequisite, thereby allowing all students to address interesting problems and explore important mathematical ideas.

As an illustration of the power of mathematical connections, consider the problem of estimating the area of a region under a curve.

Approximate the area of the region under the curve y = 2^x, above the x-axis, and between the lines x = 1 and x = 3.

How could this area be determined? One way all students could estimate the area is by determining the area of trapezoid ABCD (fig. 4.3).

Fig. 4.3. Trapezoidal estimate of the area under a curve

Students could then subdivide the interval [1,3] into two, three, . . . sub-intervals and use similar geometric reasoning to obtain better approximations of the area by using two trapezoids, three trapezoids, and so forth. Of course, a calculator or computer would be used to do the computations. This development would lead to a natural discussion of limits by all students (table 4.1). College-intending students could further extend this development to foreshadow the study of integral calculus.

                TABLE 4.1



                Refined Estimates of the Area under the Curve



                No. of Trapezoid        Area Estimate



                      1                       10



                      2                       9



                      3                       8.810



                      4                       8.743



                      5                       8.712



                     10                       8.670



                (Note: The actual area is 6/ln2 or approximately 8.656 17.)

A completely different approach is available to all students by using computer simulation to generate random points contained within the rectangle ABCF (fig. 4.4). This method is based on a probabilistic model that assumes that as the number of randomly generated points increases, the ratio on the right-hand side of the following equation will more closely approximate that on the left-hand side.

Fig. 4.4. Probabilistic model for estimating area under the curve

     Area under curve            Total no. of points under or on curve



   ------------------           --------------------------------------



     Area of ABCF                  Total no. of points

Table 4.2 reports estimates of the area under the curve for increasing numbers of computer-generated points.

This example highlights only some of the many connections among algebra, geometry, probability, and fundamental ideas of calculus.

        TABLE 4.2



        Probabilistic Estimates of Area under the Curve



        No. of Pts.       No. Pts. under Curve      Area Estimate



               10                   6                   9.6



              100                  54                   8.64



            1 000                 548                   8.676 8



           10 000               5 429                   8.686 4

College-intending students should understand and be able to use connections among mathematical topics, including the following:

• Relations and functions
• Systems of equations and matrices
• Function equations expressed in standardized form and geometric transformations
• Complex numbers represented as a + bi or r (cos + i sin ) and as ordered pairs (a,b) or (r, in the complex plane
• Right-triangle ratios, trigonometric functions, and circular functions
• Circular functions and series
• Rectangular coordinates and polar coordinates
• Algorithms and their computer implementation
• Explicit and parametric representation of equations
• A function and its inverse, such as the logarithmic and exponential functions
• Statistical procedures and their requisite probability concepts
• The conversion of data to z-scores and geometric transformations
• Finite graphs and matrices
• Recursive and closed-form definitions of the same sequence

Instruction that focuses on networks of mathematical ideas rather than solely on the nodes of the networks in isolation will serve to instill in students an understanding of, and appreciation for, both the power and the beauty of mathematics. Developing mathematics as an integrated whole also serves to increase the potential for retention and transfer of mathematical ideas. Connecting mathematics with other disciplines and with daily affairs underscores the utility of the subject.