OVERVIEW

This section presents fourteen curriculum standards for grades 9-12:

1. Mathematics as Problem Solving
2. Mathematics as Communication
3. Mathematics as Reasoning
4. Mathematical Connections
5. Algebra
6. Functions
7. Geometry from a Synthetic Perspective
8. Geometry from an Algebraic Perspective
9. Trigonometry
10. Statistics
11. Probability
12. Discrete Mathematics
13. Conceptual Underpinnings of Calculus
14. Mathematical Structure

Background

Historically, the purposes of secondary school mathematics have been to provide students with opportunities to acquire the mathematical knowledge, skills, and modes of thought needed for daily life and effective citizenship, to prepare students for occupations that do not require formal study after graduation, and to prepare students for postsecondary education, particularly college. The Standards' Introduction describes a vision of school mathematics in which these purposes are embedded in a context that is both broader and more consistent with accelerating changes in today's society. High school graduates during the remainder of this century can expect to have four or more career changes. To develop the requisite adaptability, high school mathematics instruction must adopt broader goals for all students. It must provide experiences that encourage and enable students to value mathematics, gain confidence in their own mathematical ability, become mathematical problem solvers, communicate mathematically, and reason mathematically. The fourteen standards for grades 9-12 establish a framework for a core curriculum that reflects the needs of all students, explicitly recognizing that they will spend their adult lives in a society increasingly dominated by technology and quantitative methods.

In view of existing disparities in educational opportunity in mathematics and the increasing necessity that all individuals have options for further education and alternative careers, each standard identifies the mathematical content or processes and the associated student activities that should be included in the curriculum for all students. As suggested by figure 1, the core curriculum is intended to provide a common body of mathematical ideas accessible to all students. We recognize that students entering high school differ in many ways, including mathematical achievement, but we believe these differences are best addressed by enrichment and extensions of the proposed content rather than by deletions. The mathematics curriculum must set high, but reasonable, expectations for all students.

Fig. 1. A differentiated core curriculum

The core curriculum can be extended in a variety of ways to meet the needs, interests, and performance levels of individual students or groups of students. To illustrate, many of the standards also specify topics that should be studied by college-intending students. We use the term college-intending not in an exclusionary sense, but only as a means by which to identify the additional mathematical topics that should be studied by students who plan to attend college. In fact, we believe that these additional curricular topics should be studied by all students who have demonstrated interest and achievement in mathematics.

A school curriculum in line with these standards should be organized so as to permit all students to progress as far into the mathematics proposed here as their achievement with the topic allows. In particular, students with exceptional mathematical talent who advance through the material more quickly than others may continue to college-level work in the mathematical sciences. However, we strongly recommend against acceleration that either omits content identified in these standards or advances students through it superficially.

Figure 1 also is intended to portray an expectation that mathematical ideas will grow and deepen as students progress through the curriculum and that the consolidation of learning is essential for all students during the senior year. Such a synthesis of mathematical knowledge will enhance students' prospects for securing employment and for both entering and successfully completing collegiate programs. It is, therefore, an underpinning of the proposed curriculum.

Underlying Assumptions

The standards for grades 9-12 are based on the following assumptions:

• Students entering grade 9 will have experienced mathematics in the context of the broad, rich curriculum outlined in the K-8 standards.
• The level of computational proficiency suggested in the K-8 standards will be expected of all students; however, no student will be denied access to the study of mathematics in grades 9-12 because of a lack of computational facility.
• Although arithmetic computation will not be a direct object of study in grades 9-12, number and operation sense, estimation skills, and the ability to judge the reasonableness of results will be strengthened in the context of applications and problem solving, including those situations dealing with issues of scientific computation.
• Scientific calculators with graphing capabilities will be available to all students at all times.
• A computer will be available at all times in every classroom for demonstration purposes, and all students will have access to computers for individual and group work.
• At least three years of mathematical study will be required of all secondary school students.
• These three years of mathematical study will revolve around a core curriculum differentiated by the depth and breadth of the treatment of topics and by the nature of applications.
• Four years of mathematical study will be required of all college-intending students.
• These four years of mathematical study will revolve around a broadened curriculum that includes extensions of the core topics and for which calculus is no longer viewed as the capstone experience.
• All students will study appropriate mathematics during their senior year.

Features of the Mathematics Content

Initially, it may appear that an excessive amount of curriculum content is described in the 9-12 standards. When this content is evaluated, however, it should be remembered that the proposed 5-8 curriculum will enable students to enter high school with substantial gains in their conceptual and procedural understandings of algebra, in their knowledge of geometric concepts and relationships, and in their familiarity with informal, but conceptually based, methods for dealing with data and situations involving uncertainty. Moreover, additional instructional time can be gained by organizing the curriculum so that student learning is systematically maintained and review is embedded in the context of new topics or problem situations. With these conditions satisfied, it is our belief that it will be possible to address the recommended content within a three- or four-year sequence with the expectation of a reasonable level of student proficiency.

Traditional topics of algebra, geometry, trigonometry, and functions remain important components of the secondary school mathematics curriculum. However, the 9-12 standards call for a shift in emphasis from a curriculum dominated by memorization of isolated facts and procedures and by proficiency with paper-and-pencil skills to one that emphasizes conceptual understandings, multiple representations and connections, mathematical modeling, and mathematical problem solving. The integration of ideas from algebra and geometry is particularly strong, with graphical representation playing an important connecting role. Thus, frequent reference to graphing utilities will be found throughout these standards; by this we mean a computer with appropriate graphing software or a graphing calculator. In addition, topics from statistics, probability, and discrete mathematics are elevated to a more central position in the curriculum for all students. Specific topics that should be given either increased or reduced emphasis are summarized in the chart.

SUMMARY OF CHANGES IN CONTENT AND EMPHASES IN 9-12 MATHEMATICS

TOPICS TO RECEIVE INCREASED ATTENTION

ALGEBRA

• The use of real-world problems to motivate and apply theory
• The use of computer utilities to develop conceptual understanding
• Computer-based methods such as successive approximations and graphing utilities for solving equations and inequalities
• The structure of number systems
• Matrices and their applications

GEOMETRY

• Integration across topics at all grade levels
• Coordinate and transformation approaches
• The development of short sequences of theorems
• Deductive arguments expressed orally and in sentence or paragraph form
• Computer-based explorations of 2-D and 3-D figures
• Three-dimensional geometry
• Real-world applications and modeling

TRIGONOMETRY

• The use of appropriate scientific calculators
• Realistic applications and modeling
• Connections among the right triangle ratios, trigonometric functions, and circular functions
• The use of graphing utilities for solving equations and inequalities

FUNCTIONS

• Integration across topics at all grade levels
• The connections among a problem situation, its model as a function in symbolic form, and the graph of that function
• Function equations expressed in standardized form as checks on the reasonableness of graphs produced by graphing utilities
• Functions that are constructed as models of real-world problems

STATISTICS

PROBABILITY

DISCRETE MATHEMATICS

TOPICS TO RECEIVE DECREASED ATTENTION

ALGEBRA

• Word problems by type, such as coin, digit, and work
• The simplification of radical expressions
• The use of factoring to solve equations and to simplify rational expressions
• Operations with rational expressions
• Paper-and-pencil graphing of equations by point plotting
• Logarithm calculations using tables and interpolation
• The solution of systems of equations using determinants
• Conic sections

GEOMETRY

• Euclidean geometry as a complete axiomatic system
• Proofs of incidence and betweenness theorems
• Geometry from a synthetic viewpoint
• Two-column proofs
• Inscribed and circumscribed polygons
• Theorems for circles involving segment ratios
• Analytic geometry as a separate course

TRIGONOMETRY

• The verification of complex identities
• Numerical applications of sum, difference, double-angle, and half-angle identities
• Calculations using tables and interpolation
• Paper-and-pencil solutions of trigonometric equations

FUNCTIONS

• Paper-and-pencil evaluation
• The graphing of functions by hand using tables of values
• Formulas given as models of real-world problems
• The expression of function equations in standardized form in order to graph them
• Treatment as a separate course

Patterns of Instruction

The broadened view of mathematics described in the Introduction to this document under the rubric mathematical power, together with the capabilities of available and emerging technology, suggests a need for changes in instructional patterns and in the roles of both teachers and students.

A variety of instructional methods should be used in classrooms in order to cultivate students' abilities to investigate, to make sense of, and to construct meanings from new situations; to make and provide arguments for conjectures; and to use a flexible set of strategies to solve problems from both within and outside mathematics. In addition to traditional teacher demonstrations and teacher-led discussions, greater opportunities should be provided for small-group work, individual explorations, peer instruction, and whole-class discussions in which the teacher serves as a moderator.

These alternative methods of instruction will require the teacher's role to shift from dispensing information to facilitating learning, from that of director to that of catalyst and coach. The introduction of new topics and most subsumed objectives should, whenever possible, be embedded in problem situations posed in an environment that encourages students to explore, formulate and test conjectures, prove generalizations, and discuss and apply the results of their investigations. Such an instructional setting enables students to approach the learning of mathematics both creatively and independently and thereby strengthen their confidence and skill in doing mathematics.

The role of students in the learning process in grades 9-12 should shift in preparation for their entrance into the work force or higher education. Experiences designed to foster continued intellectual curiosity and increasing independence should encourage students to become self-directed learners who routinely engage in constructing, symbolizing, applying, and generalizing mathematical ideas. Such experiences are essential in order for students to develop the capability for their own lifelong learning and to internalize the view that mathematics is a process, a body of knowledge, and a human creation.

The use of technology in instruction should further alter both the teaching and the learning of mathematics. Computer software can be used effectively for class demonstrations and independently by students to explore additional examples, perform independent investigations, generate and summarize data as part of a project, or complete assignments. Calculators and computers with appropriate software transform the mathematics classroom into a laboratory much like the environment in many science classes, where students use technology to investigate, conjecture, and verify their findings. In this setting, the teacher encourages experimentation and provides opportunities for students to summarize ideas and establish connections with previously studied topics.

The most fundamental consequence of changes in patterns of instruction in response to technology-rich classroom environments is the emergence of a new classroom dynamic in which teachers and students become natural partners in developing mathematical ideas and solving mathematical problems.

Assessment of student learning should be viewed as an integral part of instruction and should be aligned with key aspects of instruction, such as the use of technology. The reader is encouraged to examine the Evaluation standards for more detail on student assessment.

The following chart summarizes the major changes in patterns of instruction proposed for grades 9-12.

SUMMARY OF CHANGES IN INSTRUCTIONAL PRACTICES IN 9-12 MATHEMATICS

INCREASED ATTENTION to--         DECREASED ATTENTION to--







* The active involvement of stu-        * Teacher and text as exclusive



  dents in constructing and ap-           sources of knowledge



  plying mathematical idea



                                       		 * Rote memorization of facts



* Problem solving as a means as           and procedures



  well as a goal of instruction



                                    		    * Extended periods of individual



* Effective questioning tech-             seatwork practicing routine



  niques that promote                     student tasks



  interaction



                                     		   * Instruction by teacher exposi-



* The use of a variety of in              tion



  structional formats (small



  groups, individual explorations,      * Paper-and-pencil manipulative



  peer instruction, whole-class           skill work



  discussions, project work)



                                   			     * The relegation of testing to an



* The use of calculators and              adjunct role with the sole pur-



  computers as tools for learn-           pose of assigning grades



  ing and doing mathematics







* Student communication of



  mathematical ideas orally and



  in writing







* The establishment and applica-



  tion of the interrelatedness of



  mathematical topics







* The systematic maintenance



  of student learnings and



  embedding review in the con-



  text of new topics and problem



  situations







* The assessment of learning as



  an integral part of instruction

The Core Curriculum

The core curriculum for all students is the most fundamental change proposed for grades 9-12. It is very important for the reader of these standards to understand (1) exactly what is and is not being proposed, (2) the advantages of the core curriculum over current practice, and (3) the implications for teaching as decision making.

What is and what is not being proposed. For emphasis, we repeat the core-curriculum assumption here (slightly modified to highlight its significant characteristics):

[These] three years of [required] mathematical study will revolve around a core curriculum differentiated by the depth and breadth of the treatment of topics and by the nature of applications.

This assumption proposes that the curriculum topics described in this document apply to all students--except where the topics are specifically differentiated for those who are college intending. This means that the longstanding practice of requiring lower-achieving high school students to repeat sixth-grade mathematics content over and over will be replaced by a study of content that we believe provides these students, as well as their classmates, with a central core of mathematical representation, mathematical processing, mathematical problem solving, and mathematical thinking. In particular, this statement asserts that if the sequence of courses often designated as "general mathematics" does not address the content and associated goals of the core curriculum, it is no longer acceptable.

It is important to understand that this statement does not imply that students of all performance levels must be taught in the same classroom, and it does not imply that the content presentation for all students must be the same. However, no matter how individual school districts or schools choose to deal organizationally with students who exhibit different abilities, achievement levels, and interests, it is crucial that all students experience the full range of topics proposed for the core curriculum. It is equally important to ensure that the instructional practices and resources described in the previous section are integral to the mathematical experiences of all students.

Advantages this proposal offers to both students and teachers. The core curriculum proposed here offers several advantages over current practice, which dichotomizes secondary school mathematics into programs for college-bound and non-college-bound students.

Advantage 1: The core curriculum provides equal access and opportunity to all students. Mathematical literacy is vital to every individual's meaningful and productive life. The mathematical abilities needed for everyday life and for effective citizenship have changed dramatically over the last decade and are no longer provided by a computation-based general mathematics program. By removing the "computational gate" to the study of high school mathematics and recognizing that there frequently is not a strict hierarchy among the proposed mathematics topics at this level, we are able to afford all students more opportunities to fulfill their mathematical potential and participate throughout their lives as productive members of our society.

Advantage 2: The core curriculum provides greater flexibility for individual students, thus allowing them to keep their options open. Students' interests, goals, and achievements are not static but change as they mature and advance through high school. In choosing not to trap students in one of the two conventional linear patterns, we ensure that doors to college programs and vocational training are kept open for all students.

Advantage 3: The core curriculum better prepares non-college-intending students for the world of today and tomorrow. Henry Pollak's summary of the types of expectations stated by today's employers, coupled with the report Workforce 2000: Work and Workers for the Twenty-first Century (Johnston and Packer 1987), makes it apparent that much higher mathematics, language, and reasoning capabilities will be required of employees. The ever-increasing role of technology in our society further argues for a curriculum that moves all students beyond computation.

Advantage 4: The core curriculum provides opportunities for all students to confront more interesting and important mathematics. By assigning computational algorithms to calculator or computer processing, this curriculum seeks not only to move students forward but to capture their interest. As a result, students no longer will be confronted with the demeaning prospect of studying for the third, fourth, or fifth time the same content topics as their twelve-year-old siblings. We believe the opportunity to study mathematics that is more interesting and useful and not characterized as remedial will enhance students' self-concepts as well as their attitudes toward, and interest in, mathematics. These attitudinal shifts, coupled with the changes in mathematical content, should in turn provide a more interesting and stimulating environment for teaching.

In summary, the core curriculum seeks to provide a fresh approach to mathematics for all students--one that builds on what students can do rather than on what they cannot do.

Implications for teaching as decision making. In most schools, complete implementation of the 9-12 curriculum standards will require a transition period. Initially, it is likely that few students will have had the kind of mathematics experiences outlined in the K-8 standards, and teachers may need to provide some of the learning experiences described in the 5-8 standards as prerequisites to the proposed 9-12 curriculum. The amount of instructional time spent on informal activities will vary and will depend on the maturity of the students, their prior experience with a topic, and the complexity of the topic itself. In general, the relative maturity of high school students will enable them to progress more rapidly through certain topics than middle grade students could. Nevertheless, the importance of informal experiences for developing primitive conceptual understanding, a prerequisite to students' formal study and abstraction of mathematical ideas, should be recognized. In most situations, new mathematical ideas should continue to be introduced at the concrete level.

Once the prerequisite conditions have been met, we believe that a substantial amount of content in each topic area will be accessible to all students. This is not to imply that all students will experience the same coverage of each topic, but rather that the range of content topics is open to all. Often, but not always, the depth to which a topic is explored will relate to the level of abstraction at which students are capable of operating. Concrete examples and applications of a topic should be open to all; higher levels of abstraction and generalization should be available to, but not required of, all. Decisions about appropriate depth of treatment for a particular topic are a matter of teacher judgment, but we urge teachers to challenge students as much as possible. Preliminary data from other countries (Schoen 1988) developing a common curriculum for the majority of high school-aged students suggest that as teachers gain experience with these topics and the concept of content differentation, they develop effective techniques for advancing students further into the topics.

As they organize instruction for the core curriculum, it is crucial that educators differentiate between content topic and content. The Standards proposes that all students be guaranteed equal access to the same curricular topics; it does not suggest that all students should explore the content to the same depth or at the same level of formalism. The curricular topics we propose may each be further and quite naturally subdivided and their associated content developed at several levels, consistent with students' ability to abstract. It is at this level that differentiation takes place.

Examples of Content Differentiation

The following two examples suggest the kind of content differentiation on which the concept of the core curriculum is based. The levels indicated in these and other examples in the 9-12 standards should not be considered ceilings; students of varying abilities and differing needs should be encouraged and helped to progress to as high a level as they demonstrate the interest and capacity to understand. The differentiation of content, if well planned, will facilitate growth in mathematical understanding for all students.

The first example focuses on a consumer application of mathematics.

Carlos deposits $100 in a savings account earning 6% interest compounded annually. Assuming a fixed interest rate, how much money will be in the account at the end of 10 years?

Level 1: With a calculator, all students should be able to determine the amount of money in the account each year by successively applying the following relationship:

Amount at end of year = Amount at beginning of year + .06 (Amount at beginning of year).

Applying this relationship will give the following:

Amount (in dollars) at beginning of year 1 (initial deposit) = 100



Year 1:    Amount at end of year = 100 + .06(100) = 106



Year 2:    Amount at end of year = 106 + .06(106) = 112.36



Year 3:    Amount at end of year = 112.36 + .06(112.36) = 119.10



     .



     .



     .



Year 10:   Amount at end of year = 168.95 + .06(168.95) = 179.08

The use of a computer spreadsheet would be a natural extension of this activity and would give these students a powerful tool for further processing.

Students at this level also could use a pattern or template for processing:

Year 1:    Amount at end of year: 100(1.06)1



Year 2:    Amount at end of year: 100(1.06)2



     .



     .



     .



Year 10:   Amount at end of year: 100(1.06)10

They could verify this template by checking its results against corresponding values obtained by direct calculation. A discussion of how this template could be modified under different initial conditions would instill in the students further mathematical power. For example:

If the time is changed to 20 years,       the final amount = 100(1.06)20



If the starting amount is $200,           the final amount = 200(1.06)10.



If the annual interest rate is changed



to 12%,                                   the final amount = 100(1.12)10



 

Level 2: After completing the level 1 activities, students could generalize the results of their initial year-by-year calculations and use appropriate notation to assist in this process. Except for the use of technology to aid in processing, their approach as illustrated below would be traditional.

They could modify the original year-by-year table to be

or

Similarly,

.

.

.

Equation(1)

Combining

.

.

.

and in general,

where n is the number of years.

Further generalizing would give

Equation (2)

where is the initial deposit, r is the annual interest rate, n is the number of years, and is the accumulated amount in the account after n years.

Level 3: After completing level 2, these students would further generalize equation (2) where r becomes the interest rate per interest period and n is the number of interest periods. This extension would allow students to explore problems in which the annual interest rate is compounded semiannually, quarterly, monthly, daily, and so forth.

It is important to note that all students at levels 1, 2, and 3 would use calculators or computers to address related problems such as the following and discuss their results. (1) Does the amount double if (a) the interest rate is doubled or (b) the time period is doubled? (2) What are the doubling periods for amounts invested at 7%, 10%, and 20%? (3) Some people use the rule of 72, d = 72/100i, to approximate the doubling period, d, for interest rate, i. Test the formula for different interest rates and report on its accuracy. (4) How much money would you need to invest today to have $10 000 in 20 years? The solution methods for these problems could involve guessing and checking, generating a table by calculator or spreadsheet, graphing, and applying the compound interest formula. In each situation, lower-achieving students would reach their conclusions through additional numerical exercises.

Level 4: After completing level 3, students would, given three of the four variables in equation (2), solve for the fourth (e.g., ).

Students at this level might also explore the derivation of the rule of 72 by solving equation (2) using natural logarithms.

A variety of further extensions are possible. Equation (2) could be generalized to problems in which compound growth provides an appropriate representation (e.g., in biology). Earlier results could be proved by mathematical induction. Instantaneous compounding could be investigated, which would in turn lead to the development and use of the constant e.

As another example of how the content identified for the core curriculum can be differentiated in both the depth and formalism of treatment, consider the topic of mathematical modeling called for in the standard on problem solving. The reader should note that this particular activity would occur at a point in the curriculum somewhat later than the preceding one. Thus, a particular level in this example does not necessarily presume the same prerequisite understandings and degree of mathematical maturity as the corresponding level in the previous example.

A container manufacturing company has been contracted to design and manufacture cylindrical cans for fruit juice. The volume of each can is to be 0.946 liters. In order to minimize production costs, the company wishes to design a can that requires the smallest amount of material possible. What should the dimensions of the can be?

At each level the development of the mathematical model should include opportunities for class discussion regarding the simplification of the problem in terms of the thickness of the material and wasted material when components are cut. Next, all students would derive (as necessary) formulas for the volume (V = r ²h) and surface area [S = 2r (h + r)] of a cylinder and then form cylinders of various dimensions (fig. 2), comparing their volumes and surface areas.

Fig. 2. Cylinder components

Level 1: Students should first note that since the volume is to be 0.946 liters, or 946 ml, and since 1 ml = 1 cm³, the dimensions of the can can be most easily expressed in centimeters. Rewriting the formula V = 946 = r ² h as h = 946/(r ²) permits the height to be found once a value for r is specified. Now students can use a calculator to build a table of values (table 1) consisting of heights and areas corresponding to chosen radii. The table would be analyzed for an approximate solution (radius between 5 cm and 6 cm with corresponding heights between 12.0 cm and 8.4 cm).

      TABLE 1



      Cylinder Data



      Radius (r) cm     Height(h)cm    Surface Area (S) cm2



          1                       301.1                    11898.3



          2                        75.3                      971.1



          3                        33.3                      687.2



          4                        18.8                      573.5



          5                        12.0                      535.5



          6                         8.4                      541.5



          7                         6.1                      578.2



          8                         4.7                      638.6



          9                         3.7                      719.2



         10                         3.0                      817.5

Level 2: Students would develop the algebraic expressions as in level 1 and design an algorithm to produce appropriate values. One possible algorithm is shown below.

For r = 0.5, 1, 1.5, ......, 10



    h  946/(3.14159 · r 2)



    S  2(3.14159) · r(h + r)



Output, r, h, S

Following a computer implementation of the algorithm, students would analyze the given output (table 2). This would suggest that the optimal length for the radius is again between 5 cm and 6 cm. However, the advantage of using computer methods is that students can easily approximate this length to the nearest 0.1 cm by simply modifying the program so the loop runs between 5 and 6 in increments of 0.1. A run of the modified program yields r is approximately equal to 5.3 cm and h is approximately equal to 10.7 cm.

      TABLE 2



      Refined Cylinder Data



      Radius (r) cm     Height (h) cm   Surface Area (S) cm squared











         0.5                 1204.485 63                3785.570 8



         1                    301.121 407               1898.283 18



         1.5                  133.831 736               1275.470 49



         2                     75.280 351 7              971.132 72



         2.5                   48.179 425 1              796.069 875



         3                     33.457 934 1              687.215 287



         3.5                   24.581 339 3              617.540 384



         4                     18.820 087 9              573.530 88



         4.5                   14.870 192 9              547.678 84



         5                     12.044 856 3              535.479 5



         5.5                    9.954 426 67             534.066 195



         6                      8.364 483 52             541.527 813



         6.5                    7.127 133 88             556.541 279



         7                      6.145 334 83             578.161 534



         7.5                    5.353 269 45             605.695 542



         8                      4.705 021 98             638.623 52



         8.5                    4.167 770 33             676.547 991



         9                      3.717 548 23             719.159 803



         9.5                    3.336 525 28             766.214 89



         10                     3.011 214 07             817.518 001

Level 3: At this level, students would use the specified volume and the formulas for volume and surface area to express the surface area as a function of the radius. A graphing utility would be used to plot the function, and the graph (figs. 3 and 4) would be examined to determine the minimum surface area, S, and corresponding radius, r, from which the corresponding height, h, can be determined.

Fig. 3. Graph of algebraic representation

Fig. 4. Magnification showing that the minimum value of S is about 533.47 when r is about 5.32 (in this instance, h 10.64)

Level 4: After completing either the level 2 or level 3 activity, students would use different volume values in an attempt to discover a general relationship between radius and height that would minimize the surface area for a given volume. The analysis of accumulated data from several trials will suggest that the surface area appears to be minimal when h is twice r.

Level 5: Students at this level would prove that the surface area is minimal when h = 2r. This task should be viewed as an open-ended project, since it would not only require the generalization of the more familiar arithmetic-geometric mean inequality for two numbers to that for three numbers [(a + b + c)/3 the cube of the square root of abc and the equality holds if and only if a = b = c] but would also entail ingenuity and creative mathematical thought.

One approach is to observe that since

                   S = 2r(h + r) = 2rh + 2r 2







is to be minimized and







                   2rh + 2r 2 = rh + rh + 2r 2,







it follows that







                   rh + rh + 2r 2



                   -----------------   323r4 h 2



                            3







or







                           rh + rh + 2r 2    3 323r4h2.

The left-hand side is minimal when the equality holds; that is, only if rh = rh = 2r ². It follows that the surface area is minimal when rh = 2r ², or h = 2r.

The levels of topic differentiation in the preceding two examples should not be viewed as prescriptive but rather as suggestive of the range of possible levels of treatment by which all students could be provided an opportunity to learn important mathematical ideas.

Additional examples of how topics in the core curriculum may be treated at differing levels of abstraction can be found in the discussion sections of the standards on algebra, trigonometry, and probability. When interpreting the examples of content differentiation, it is important to understand that all students are not expected to progress from the informal work usually associated with levels 1 or 2 to the advanced work associated with levels 4 or 5. Some students may begin and complete the consideration of a topic or problem at level 1 and then revisit the topic or problem later in the curriculum at a higher level, consistent with their growing understanding of mathematics. Others may begin the consideration of a topic at level 1 and then progress to work at a higher level within the course of a single unit. Depending on their interests and performance, a subset of this group of students will consider the topic in its full and rich detail at an appropriate point in the curriculum.

Summary