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Part Three—
Curricular Considerations

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Overview
Any discussion of curriculum assumes, whether implicitly or explicitly, one of many views of curriculum. College admissions requirements, for example, sometimes describe a high school mathematics curriculum as little more than a list of course titles. Toward the other end of specificity, some might point to a textbook, looking especially at its table of contents. For the purposes of this document, a curriculum is a detailed plan for instruction, including not only the materials used by teachers and students, but also understandings of how they fit together and of the important mathematical concepts embedded within them. Thus, a mathematics curriculum might include—but must not be limited to—tasks such as those in this volume. The essays in Part Three in particular go beyond a list of topics, beyond a collection of tasks, to discuss how curricula might be constructed—how tasks, topics, and ''habits of mind" might be knit together as themes or strands to create a curriculum with coherence, depth, and rich opportunity for student learning, sense making, and connections to students' ways of thinking about the world.
In his essay, Zalman Usiskin discusses the importance of applying knowledge to new situations. When designing curriculum, he begins with a key concept and seeks models—settings, often from the workplace or everyday life, which embody the mathematical ideas. Of course, for many users of mathematics,

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it is more common to go in the other direction—to begin with a real-world setting and seek a mathematical model of that setting, perhaps with graphs or formulas. Chazan and Bethell (p. 35) describe such an approach earlier in this volume. They put students into the workplace and asked them to find the mathematics. Thus, whether the mathematics models the world or the world models the mathematics depends upon where you begin. Ultimately, both directions might be necessary for students to make strong connections between mathematics and their worlds. In discussing the role of tasks in curriculum, Usiskin suggests organizing curriculum around sequences of such models and problems that range over many years.
Albert Cuoco takes another approach to organizing curriculum. He suggests that mathematics is more than a collection of topics organized under broad headings such as geometry and number but, rather, is about ways of thinking, or habits of mind, such as algorithmic thinking, proportional reasoning, and reasoning through thought experiment. Habits of mind, he suggests, can be threads that help organize curriculum, for without habits of mind, higher order skills will remain elusive.
Harvey Keynes' essay discusses a key theme in this document: effectively preparing students for work and for higher education. Keynes asks two important questions: First, what are characteristics of tasks that can prepare students both for the workplace and for post-secondary education? Second, what are the requirements for effective use of tasks like these in the classroom? Keynes suggests conditions of appropriateness for tasks that can develop both concrete and abstract thinking.
The tasks in Part Three differ from those in other parts to illustrate that they may be fruitfully approached at many different times in a student's mathematical career. The discussions of these tasks include multiple solutions, many extensions, and more connections to other mathematical ways of thinking. Some of this discussion is quite deep mathematically, not to suggest that such depth is appropriate for all students at the same time, but, rather, to suggest that these tasks can provide opportunity for engagement in rich and deep mathematics to students when they are interested and ready. Tasks that are sufficiently rich and that satisfy Keynes' conditions for appropriateness can fit more than once into a curriculum that is organized around Cuoco's "habits of mind" or Usiskin's sequences of models.
The Lottery Winnings (p. 111) task may be solved at one level with spreadsheets, for students having little formal algebraic experience, and it may be used to motivate students to see a need for the general symbolic language that algebra provides. At another level, students who are more experienced with the symbolism of algebra might be expected to express the task's spreadsheet relationships in standard algebraic notation. Students with even more sophistication might be expected to find the general formula. In a precalculus course, students might explore these lottery winnings with annual,

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semi-annual, monthly, weekly, and daily payments as excellent background for the important calculus idea of successive approximation.
The other tasks in this part are similarly rich. In addition to discussion of probability and Simpson's paradox, Hospital Quality (p. 115) can lead to exploration of ideas about rates, ordered pairs, and vectors. Rounding Off (p. 119) provides opportunities for exploring ideas in algebra and arithmetic, while serving as an introduction to the idea of using geometry to represent probability. Rules of Thumb (p. 123) can lead to discussion of modeling and comparison of linear models, or linear and quadratic models. The many avenues of approach to these tasks may be exploited by teachers to maximize connections to students' thinking and experience.
The idea of periodically revisiting tasks sounds rather like Bruner's spiral curriculum (Bruner, 1965/1960), an idea that some might argue has lost its usefulness. Data from the Third International Mathematics and Science Study (TIMSS) (Schmidt, McKnight, & Raizen, 1996) suggest, however, that there has been a degeneration of the spiral curriculum as Bruner saw it. The point of Bruner's spiral curriculum was not that topics should be repeated for several years until they "stick" but that, when an idea is revisited in a new setting or with new tools, if students have opportunity to connect the new encounter to their understandings of their previous encounters with the idea (along with all the intervening experiences), then their understanding can grow.
There are dangers in any statements of standards or suggested visions of school mathematics, for there is no clear path indicating what should happen in classrooms. Cuoco warns that the statement, "Students should be able to solve problems like these," can become "Students should be able to solve these problems." But such a conclusion is contrary to the intention of this volume. The goals are for students to learn mathematics and to learn to appreciate the power that mathematics holds for us. Any task or collection of tasks is merely intended to be a means to those ends. Furthermore, teaching any task as only a procedure to be memorized will destroy its richness. Our point—especially in evoking the image of the spiral curriculum—is that there is value in revisiting tasks such as these at various points in a student's career, each time aiming for more sophisticated analysis and deeper mathematics.
In summary, the tasks in this volume cannot comprise a high school mathematics curriculum; no small collection of tasks could. These tasks have been chosen for their illustrative richness rather than for any collective curricular coherence. Individually and collectively, these tasks together with the essays might instead serve as inspiration for those interested in curriculum, but, as curriculum designers know, there is a lot of work to be done between first noting that there is mathematics in some real-world context and finally developing good curricular materials.

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References
Bruner, J. S. (1965/1960). The process of education. Cambridge, MA: Harvard University Press.
Schmidt, W. H., McKnight, C. C., & Raizen, S. A. (1996). A splintered vision: An investigation of U.S. science and mathematics education. Dordrecht, The Netherlands: Kluwer Academic Publishers.

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10—
Fitting Tasks to Curriculum
ZALMAN USISKIN
University of Chicago
When I first taught high school, I used to tell my students—even the average ones—that the real test of learning was not whether they could answer questions like those they had seen in their textbooks but whether they could apply their knowledge to new situations they had not encountered. This aphorism is only partially true and was patently unfair. In applying the principle of the aphorism, when I would make up a test, I would purposely choose items that students had not encountered, items for which they would not have studied. Those items were not a test of what had been learned from the class but what had not been learned from the class. They tested some natural or acquired competence beyond the course.
Those who wish students to apply, synthesize, analyze, and evaluate (to use the language of higher mental processes found in Bloom's Taxonomy of Educational Objectives [1956]) have always found it difficult to invent representative items. Those for whom a problem is "a situation which we want to resolve but for which we do not have an algorithm" (to use the common researcher definition) have a similar dilemma, for once a problem is solved, the astute solver has an algorithm to use for the next problem of that type. Inventing good problems has always been an art.
The quandary presented by the desire to have students apply their knowledge

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and not just parrot it has been felt by all those whose goals involve more than routine skills. In the 1970's, when in a reaction to one of the weaknesses of the "new math" we began to design curricula in which a main goal was to have students apply what they knew in real-world situations, the same dilemma appeared in only slightly different clothing. We felt strongly that students were not able to apply algebra because they were not taught the applications. But if we taught the applications, then were we not changing "application" from a higher level process to a lower one?
We decided that the goal of learning to apply was more important than how that learning had been attained; that is, we decided to teach the applications. For example, consider the following problem, introduced in Algebra Through Applications with Probability and Statistics (Usiskin, 1979).
In Chicago there are two monthly rates for local telephone service. Choice 1 has a base rate of $11.25 for 200 calls plus .0523 for each call over 200. Choice 2 is $24.50 for an unlimited number of calls. How do you decide which plan is better?
Students were asked to write a sentence that would help them decide. The goal was to think of the sentence 11.25 + .0523(x - 200) > 24.5 (When is choice 1 better?) or 11.25 + .0523(x - 200) < 24.5 (When is choice 2 better?). This is not an easy task for students who have never studied problems like these. But we wanted to make solving such problems routine because they abound in the real world. The lesson contained similar items involving teacher salaries (compare $9,000 plus $500 for each year's experience with $9,750 plus $350 for each year) and rental cars (compare $15.95 a day plus 14¢ a mile with $12.95 a day plus 15¢ a mile). Fitting the title of the lesson, "Decision-Making Using Sentences," students were not asked to solve the sentences they wrote. The problems were employed to motivate the next lesson, in which students were shown an algorithm for solving ax + b < cx + d, and were given additional problems of the type.
"Problems of the type" is an important phrase to consider. What type is involved here? A current view is that it is unwise to sort problems by their context, such as has been the tradition in algebra with coin problems, mixture problems, distance-rate-time problems, age problems, and so on. Yet, on the other hand, Polya's advice is also commonly accepted: "If you cannot solve the proposed problem try to solve first some related problem" (Polya, 1957). When is a problem to be considered as "related''? How should we group problems for study?
The consequences of grouping related problems reach far beyond explication of types. With respect to problem solving, the power of mathematics lies in its ability to solve entire classes of problems with similar techniques. The Chicago telephone-cost problem is not an earth-shaking context for mathematics, but it exemplifies a class of constant increase problems that lead to equations and functions involving the algebraic form ax + b. Put another way, if we expect

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students to come up with a mathematical model for a real situation, they need to know the attributes of the situation that would cause a particular mathematical model (linear, quadratic, exponential, sine wave, etc.) to be appropriate.
So, in developing the University of Chicago School Mathematics Project curricula that give strong attention to applications, we have often begun with the mathematical concept and sought the key mathematical models of that concept (University of Chicago School Mathematics Project, 1989-97; Usiskin, 1991). In a few instances, the content is standard in the curriculum, as with growth and decay models for exponential functions. In other cases, the mathematical conceptualizations of the topic need to be broadened; as with angle, for example, which in geometry is traditionally "the union of two rays," but which in applications may be better conceptualized as a "turn" or as a "difference in directions." Freudenthal (1983) has done many analyses of this kind.
In a few cases, we have found that the standard approach to the problem type to be inhibiting. Consider the following problem, which originates from an actual situation:
A city charges 8% tax and a restaurant in the city gives a 5% discount for paying cash. Is it better for a diner if the discount is given first and the tax charged on the discounted price, or if the tax is charged on the discounted price, and then the discount taken?
Students are customarily taught that taxes (discounts) are added to (subtracted from) original prices to determine total cost. Thinking this way, working from a meal with original cost M, the first option is represented by the expression
If, instead, students are taught to think of taxes and discounts as factors, i.e., to think multiplicatively, that same option is represented by 1.08(.95M). The multiplicative representation is not only simpler but makes transparent the desired generalization from doing this sort of problem: it makes no difference what the specific discounts and taxes are; if they are fixed they can be done in any order.
Fitting tasks to curriculum involves more than assuring that the scope of the curriculum is broad enough to accommodate the tasks. There is also the question of the sequence of topics. The mathematics you will see illustrated in the Lottery Winnings (p. 111) task involves the general idea of annuities, which can be viewed as the sums of compound interest expressions, which themselves trace back to the same multiplicative idea in the restaurant example given immediately above, which in turn requires that a student have the notion that multiplication by a number larger than 1 serves to enlarge a quantity, and multiplication by a number between 0 and 1 serves to contract it.
In the past, the mathematics curriculum has been carefully sequenced either by algorithmic considerations (to perform long division, you must be able

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to subtract and multiply, so these operations must precede division) or by logical considerations (one proof of the Pythagorean Theorem involves similar triangles, so these must be studied before the Pythagorean Theorem can be considered). The above analysis suggests that the development of problem-solving among the populace would be aided by the development of sequences of models and problems that range over many years of study.
Here is an example of such a development. Begin in the primary grades with the use of subtraction for comparison and the specific example of change. When division is introduced, cover the wide range of rates such as students/class, km/hr, and people/mi2. In middle school, use negative numbers to represent measures in situations that have two opposing directions, such as gain and loss, up and down, or north and south, and picture them on the number line. Introduce ordered pairs, not only for cataloguing the locations of objects but also for recording pairs of data. Then, by asking how fast something has changed, introduce the concept of rate of change, picture this in the coordinate plane, and use both the application and the picture to lead into the idea of slope. In high school, study situations in which the rate of change is not constant. Use these to consider limits of rates of change. There is reasonable evidence that such an approach is far more effective in leading to understanding of the pure and applied mathematics involved than traditional approaches, in which the idea of slope is introduced by a definition as (y2 - y1)/(x2 - x1) with no prior buildup or connection to rate of change.
Another example is geometric. In the elementary grades, use the familiar coordinate square grid to obtain areas of rectilinear figures and associate the product xy with the area of a rectangle with dimensions x and y. But also modify the square or rectangular grid to generate tessellations. Point out that a two-dimensional object that tessellates can be cut from a large sheet without wasting space. In the middle grades or early high school, use finer and finer grids to provide better and better estimates of the areas of regions. In high school, graph the speed of a car or other object over time, and interpret the area between the graph and the x-axis as the product of the speed and the time, i.e., as the distance traveled. This paves the way for the many situations representable with integrals.
It is significant that the long sequences described in the preceding two paragraphs are embedded in the traditional content of arithmetic, algebra, geometry, and elementary analysis. We have yet, however, to develop long sequences for the teaching of statistics, as it has had a shorter lifetime in the high school curriculum. To incorporate tasks like those in this volume into the experience of students is a curricular problem that is currently being undertaken by some of the mathematics reform curricula.
Even with the analysis of individual tasks and their setting in the curriculum, there remain two particularly knotty curricular problems. First, there are tasks that involve a range of mathematics too wide to be classified by

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a single mathematical model or even a family of related models. Incorporating these tasks into a curriculum is on the one hand easy because they can fit in so many places. On the other hand, without such a broader context in which to embed them, such tasks become unwieldy if students are not well versed in the prerequisites to them.
Second is the issue with which this essay began. While a fundamental goal of mathematics education must remain for students to acquire the competencies to solve simple and complex problems they are likely to encounter in their lives, students must also have opportunities to approach problems the likes of which they have not seen before. A task for curriculum developers is to accommodate these two competing needs. The corresponding task for philosophers and policy makers is to consider whether it is fair for everyday classroom assessments to test students on the latter.
References
Bloom, B. (Ed.). (1956). Taxonomy of educational objectives. Handbook I: Cognitive domain. New York: David McKay.
Freudenthal, H. (1983). Didactical phenomenology of mathematical structures. Dordrecht, The Netherlands: Kluwer Academic Publishers.
Polya, G. (1957). How to solve it. (Second ed.). Princeton, NJ: Princeton University Press.
University of Chicago School Mathematics Project. (1989-97). Transition mathematics. Algebra. Geometry. Advanced Algebra. Functions, statistics, and trigonometry. Glenview, IL: Scott Foresman.
Usiskin, Z. (1979). Algebra through applications with probability and statistics. Reston, VA: National Council of Teachers of Mathematics.
Usiskin, Z. (1991). Building mathematics curricula with applications and modelling. In M. Niss, W. Blum, & I. Huntley (Eds.), Teaching of mathematical modelling and applications, (pp. 30-45). London: Ellis Horwood, Ltd.
ZALMAN USISKIN is Professor of Education at the University of Chicago and Director of the University of Chicago School Mathematics Project. He is a member of the Board of Directors of the National Council of Teachers of Mathematics, of the Mathematics/Science Standing Committee of the National Assessment of Educational Progress, and of the United States National Commission on Mathematical Instruction. He has served as a member of the Mathematical Sciences Education Board.

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numbers and trends that give rise to statistics. It will also give them a better sense of what to believe and what to question when confronted with statistical assertions.
Mathematical Analysis
To check the directors' assertions, one must compute death rates. For example, the death rate for patients in good condition at Mercy is 6/600 or 1%. The other results are shown in Table 2.
TABLE 2: Patient mortality, two hospitals, with rates
MERCY HOSPITAL
EXCELSIOR HOSPITAL
PATIENTS
DEATHS
RATE
PATIENTS
DEATHS
RATE
In Good Condition
600
6
1%
600
8
1.33%
In Poor Condition
1500
57
3.8%
200
8
4%
Combined Total
2100
63
3%
800
16
2%
Looking only at the combined death rate, it looks like Excelsior is the better hospital, for a 2% death rate is better than 3%. Looking at the separate death rates, however, the picture is different. For patients in good condition, the death rate is lower at Mercy. Similarly, a patient in poor condition is better off at Mercy. So the public relations director at Mercy is correct: Mercy Hospital has a better success rate both with patients in good health and with those in poor health. The reason Mercy loses more patients overall is that it treats many more seriously ill patients.
Here's an easy way to see how averages based on aggregates can deliver a different message than averages based on components. Suppose a company, in an attempt to recruit women into all positions, pays them more than men in all positions. If it is easy to recruit women for the low-paying positions, and hard to recruit them for the high-paying positions, it is possible that the average salary for women will still be lower than the average salary for men, seemingly contradicting the company's intent to pay women more.
Extensions
Students might find and analyze employment and salary patterns in various professions. They might look at admissions rates at a university by gender or by race, for the university as a whole, and then separated by college or by department. Such assignments should not be given, however, without allowing for discussion of equity issues that can be raised by such data.
Students might construct data that illustrates analogous paradoxes in contexts that appeal to them. In baseball, for example, it is possible for a batter to have the best batting average before the all-star break and the best average after the all-star break and yet fail to have the best average for the whole season.
Students might also explore other instances of weighted averages, perhaps first as simple ways of computing more familiar averages. For example, if a teacher explains that homework counts 50%, each of three exams count 10%, and the final exam counts 20%, a student

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can determine his or her average going into the final as follows:
The arithmetic in this task deserves comment. If one thinks of the death rates as fractions, then one might consider the relationship between the separate death rates and the combined death rate to be like addition. In the case of Mercy hospital, the "addition" looks as follows, where the ⊕ indicates that this is not the standard addition of fractions.
Notice that this "addition" is performed by adding the numerators and adding denominators—one of the mistakes that students make when they are supposed to perform the standard addition of fractions. Yet, this "addition" is used in many contexts, from computing batting averages in baseball to computing terms in Farey sequences, an advanced topic in number theory.
Students might be asked, "Why does this 'addition' make sense here?" "What is the difference between this and the standard addition of fractions?" "What is different about the contexts that gives rise to a different kind of addition?" Discussion of such questions can provide for a firmer understanding of the concepts of fraction, rate, and average.
This kind of "addition by component" is reminiscent of addition of vectors, which gives us a geometric model of the situation. The data for patients in good condition at Mercy (600 patients, 6 deaths) can be represented as the vector (600, 6) which can be represented geometrically as an arrow from the origin to the point (600, 6) on a coordinate plane. (See Figure 1.) Then the death rate, 6/600, is precisely the slope of the vector. By similarly representing the data for patients in poor condition as the vector (1500, 57), the sum of the vectors is given by adding the components of the vectors. That is, (600, 6) + (1500, 57) = (2100, 63). Geometrically, the sum of these vectors is the diagonal of the parallelogram formed by the vectors. (See Figure 1.) Note that because the death rate is represented by the slope of the vector, a steeper vector corresponds to a higher death rate. We can similarly represent the data from Excelsior Hospital (Figure 2).
FIGURE 1: Patient mortality at Mercy Hospital
FIGURE 2: Patient mortality at Excelsior Hospital

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Superimposing the data from Excelsior Hospital upon that from Mercy (Figure 3) shows that the sides of the Excelsior parallelogram are steeper than the corresponding sides of the Mercy parallelogram, but Mercy has a steeper diagonal. To gain a spatial and kinesthetic sense of this paradox, students might use dynamic geometry software to draw such a picture to construct data that exhibit this paradox.
Observe that the diagonal representing the sum must be between the two vectors, indicating that the slope of the sum must be between the other slopes. This provides a compelling geometric argument for the algebraic fact that defined as above, is always between the two fractions a/b and c/d, as long as a, b, c, and d are all positive. Proving this algebraically, on the other hand, requires some non-obvious techniques.
FIGURE 3: Patient mortality, two hospitals

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Rounding Off
Task
In a certain multi-million dollar company, Division Managers are required to submit monthly detail and summary expense reports on which the amounts are rounded, for ease of reading, to the closest $1,000. One month, a Division Manager's detail report shows $1,000 for printing and $1,000 for copying. In the summary report, the total for ''printing and copying" is listed as $3,000. When questioned about it by the Vice President, he claims that the discrepancy is merely round-off error. In subsequent months, the Vice President notices that such round-off errors seem to happen often on this Division Manager's reports. Before the Vice President asks that the Division Manager re-create the reports without rounding, she wants to know how often this should happen.
Commentary
We are often quoted rounded numbers that do not then turn out to be quite exact. Even a bank's approximate computational program for principal and interest can eventually drift far enough off the actual payment for the difference to be important. In any problem, we have to be concerned about which numbers are exact and about the accuracy of those that are not.
People don't often realize how huge the consequences of rounding numbers can be. Suppose, for example, that a company's board of directors has received a report indicating that each of the machines manufactured by their company will take up 2% of the freight capacity of their cargo planes, and the board wants to know how many machines can be shipped on each plane. In our standard notation, 2% represents a number somewhere between 1.5% and 2.5%. Solving the problem with each of these two exact percentages yields answers that are quite different. Using 1.5%, the board will find that the plane can hold 100% ÷ (1.5%/machine) = 66 machines; but by using 2.5%, the board will find that the plane can hold 100% ÷ (2.5%/machine) = 40 machines. So, in truth, all the board can say is that the answer is between 40 and 66 machines! Clearly, the report has not supplied accurate enough information, especially if the profitability of the shipment depends strongly on the number of machines that can be shipped.
If, on the other hand, the report had indicated that the board could assume another decimal place of accuracy, by stating that each machine accounted for 2.0% of the plane's capacity, then, with rounding, the board can be sure that the exact portion is somewhere between 1.95% and 2.05%. Using these exact percentages, the board can conclude that the plane can hold between 48 and 51 machines. One decimal place of additional accuracy in the reported data reduced the uncertainty in the answer from 26 machines to 3.
This problem is important for another reason as well, for its solution introduces a useful mathematical connection: the notion of geometric probability, where the range of options (technically, the "sample space") is represented by a geometric figure so that the probability of certain events correspond to the areas of certain portions of that figure. Geometric probability enables us to use our knowledge of the area (or length or volume) of geometric figures to compute probabilities.
Mathematical Analysis
Fundamental to an understanding of geometric probability is the idea that on a portion of a

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line, probability is proportional to length, and on a region in a plane, probability is proportional to area. For example, suppose that in Figure 1, the areas of regions A, B, and C are 2, 1, and 3 respectively, for a total area of 6. Then a point picked at random from these regions would have probability of 2/6, 1/6, and 3/6 of being in regions A, B, and C respectively.
FIGURE 1: An area model for probability
Note that the boundaries of the regions are not significant in the calculations because they have no area. Ideally (as opposed to in a physical model) these boundaries are lines with no thickness. Thus, the probability that a point from this rectangle will lie exactly on one of these boundaries, rather than close to a boundary, is zero.
In order to answer the question at hand, it must be stated more mathematically: Given a pair of numbers that both round to 1, and assuming that all such pairs are equally likely, find the probability that their sum rounds to 2. This assumption may or may not be reasonable in a particular business and would require some knowledge of typical expenses and some non-mathematical judgment.
A number that rounds to 1 is somewhere between .5 and 1.5. These numbers may be represented by a line segment, shown as the shaded portion of the number line in Figure 2.
FIGURE 2: A linear representation of numbers that round to 1
To state this a bit more formally, a number x will be rounded to 1 if .5 < x < 1.5. (Again, we can ignore the boundaries, 5 and 1.5, because the probability that a number will be exactly on the boundary is zero.) Suppose y also rounds to 1, so that .5 < y < 1.5. If we consider a coordinate plane with points (x, y), these two inequalities determine a square of side 1. This square (Figure 3) represents all pairs of numbers where both could be rounded to 1. For example, point A represents (.8, .6), B represents (1.1, 1.1), and C represents (1.3, 1.4).
FIGURE 3: An area representation for 1 + 1

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What can we say about x + y for points inside the square? Most of the time, x + y will round to 2, but sometimes it will round to 3, and sometimes it will round to 1. Note that the components of A add to 1.4, which rounds to 1; the components of B add to 2.3, which rounds to 2; and the components of C add to 2.7, which rounds to 3.
The probability that 1 + 1 rounds to 1 is the fraction of the square containing pairs that, when added, round to 1. Now, x + y rounds to 1 if x + y < 1.5, which will occur for points below the line x + y = 1.5. Similarly, x + y rounds to 3 for points above the line x + y = 2.5. These conditions each cut off a triangular corner of the square (shown as the darker shaded regions in Figure 4).
The legs of these right triangles are each of length 1/2, so they each have area 1/8. Thus, the probability that 1 + 1 = 3 is 1/8, and the probability that 1 + 1 = 1 is also 1/8. Finally the probability that 1 + 1 = 2 is 3/4, the remaining fraction of the square.
FIGURE 4: An area representation for 1 + 1, with rounding boundaries
Extensions
What's the probability that 1 × 1 = 2? This requires calculating the portion of the square that satisfies xy > 1.5 (Figure 5). Is this bigger or smaller than 1/8, calculated as the area of the upper triangle in Figure 4? A comparison of Figures 4 and 5 shows remarkable similarity. What is the precise relationship between the line x + y = 2.5 and the curve xy = 1.5? Solving the first equation for y and substituting into the second yields x(x - 2.5) = 1.5, a quadratic which simplifies to -x2 + 2.5x - 1.5 = 0 or 2x2 -5x + 3 = 0. This second equation factors easily as (2x - 3)(x - 1) = 0, yielding solutions x = 1.5 and x = 1. These solutions imply that the line x + y = 2.5 and the curve xy = 1.5 intersect the square at the same points. By the concavity
FIGURE 5: An area representation for 1 × 1, with rounding boundaries

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of the curve xy = 1.5, the curve must lie below the line inside the square. So the answer should be a little bigger than 1/8 = .125.
Calculus allows us to calculate the shaded area as precisely:
Similarly, if x = .6 and y = .7, then xy = .42 < .5, which would round to 0. The probability that xy rounds to 0 is .5 in 2 - .25 < .097.
What about 1/1? It rounds to 0 with probability .0625, to 1 with probability .75, to 2 with probability .175, and to 3 with probability .0125. These calculations require only geometry, no calculus.

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Rules Of Thumb
Task
Some drives learn the rule of thumb, "Follow two car lengths behind for every 10 miles per hour." Others learn, "Stay two seconds behind the car ahead." Do these two rules give the same results? Is one safer than the other? Is one better for roads with speed limits of 45 or 50 miles per hour and another for highways on which the speed limit is 65 or 70 miles per hour?
Commentary
Obtaining a driver's license has become one of the "rites of passage" in the U.S. On almost every written driver's test, applicants are asked how closely one driver should follow another on the highway. We all appreciate the dangers of tailgating—not enough stopping time and not enough space to avoid an accident. However, it is not clear that there is agreement about what actually constitutes tailgating—how far apart cars should be.
Rules of thumb are helpful guidelines—sometimes derived from experience—that are calculated using easily available measurements. Often they are developed under particular conditions and may be extremely inaccurate if those conditions are not fulfilled. The existance of two rules of thumb for the same situation suggests a natural question: Are the two rules simply two different ways of saying the same thing or are they offering different advice? As stated, the rules may provide visual images of how far to stay behind another car, but translating that understanding into practice on the road may be quite a different matter. The exercise of interpreting rules of thumb and comparing their results with real data could help students realize that the rules they use have implications for their actions. Also, there is the reality of high incidences of automobile accidents among new drivers. This exercise may help students examine and improve their driving habits.
In order to do the task, students need to know what it means to make a comparison. They have to identify the quantities needed in order to calculate the following distances given by the two rules and represent the rules mathematically. There are many ways to do this—written descriptions, tables, equations, or graphs, all basic tools of mathematical literacy. A comparison requires that the two representations use the same units of measurement—hence some conversions are necessary from the units used in the original rules of thumb. Such conversions are an essential part of many everyday situations, both at work and at home.
Mathematical Analysis
To begin, students might be well advised to consider the case in which two automobiles are traveling at a steady rate. The information presented is not complete and students will find that they have to seek out missing data. Naturally, what students seek will depend on their interpretation of the task. One necessary piece of information may be average car length.
The units for the car-length rule are miles per hour and car lengths, and the units for the two-second rule are miles per hour and seconds. To compare the two rules, both need to be written in the same units. A typical sedan is about 14 feet, so the car-length rule might be translated as "follow about 28 feet behind for every 10 miles per hour" or as the equation y = 28(x/10), where x is the speed of the car in miles per hour and y is the following distance in feet.
If a car is traveling at x mph, then it travels x miles in one hour—in other words, x/3600

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miles in one second. The two-second rule is then "if your speed is x mph, follow about 2x/3600 miles behind." As an equation, it is z = 2(x/3600), where x is again the speed of the car in miles per hour, but this time z is the following distance in miles (not feet as in the previous equation), and we use a different letter to distinguish it from y above.
Now the rules are both in terms of miles per hour and units of distance but not the same units of distance. The car-length rule is as follows:
where y is the following distance in feet. The two-second rule is
where z is the following distance in miles.
where z is the following distance in miles. Simplifying the car-length rule gives
where y is the following distance in feet. Simplifying the two-second rule gives
Now it's a matter of converting z to feet (or y to miles). There are 5,280 feet in a mile, so x/1800 miles is 5280(x/1800) feet. That's about 2.93x feet—very close to the distance given by the car-length rule!
Some driver's manuals give data on the distance cars travel before they are able to come to a complete stop. Often the distance is broken into two components, the reaction distance and the braking distance. The reaction distance is the distance traveled while the driver reacts to a situation and hits the brakes. The braking distance is the distance traveled from the time the brakes are applied until the car comes to a stop. A simplified version is given in Table 1.
TABLE 1: Reaction and braking distances for various speeds
SPEED
REACTION DISTANCE
BRAKING DISTANCE
20 mph
20 feet
20 feet
30 mph
30 feet
45 feet
40 mph
40 feet
80 feet
50 mph
50 feet
125 feet
60 mph
60 feet
180 feet
This table allows a comparison of the distances given by the rules of thumb with actual stopping distances. But the stopping distances are the distances required for a car to stop before hitting an immovable object blocking the road, whereas the rules of thumb assume that the car in front is also moving forward. This table suggests some questions about the rules of thumb: How much reaction time does each rule allow? Why are the rules of thumb linear and the stopping distances non-linear—and does this matter?
Extensions
In 1977, a National Observer article stated, "The usual rule of thumb in the real-estate business is that a family can afford a house 2 to 2 1/2 times its income." Incomes and housing prices have changed considerably since 1977, and real-

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estate agents' rules of thumb may have changed as well. Every subject—from shop to physics, from auto mechanics to economics—introduces rules of thumb that work well in appropriate situations. Even in mathematics, practices that students don't understand may acquire the status of rules of thumb for them and may be misapplied.
The original rule of thumb gave the measurement of a person's waist in terms of the measurements of their thumb, wrist, or neck. "Twice around the thumb is once around the wrist. Twice around the wrist is once around the neck. Twice around the neck is once around the waist." (The Dutch refer to "rules of fist," possibly for similar reasons.) The differences in body proportions at different ages (see Figure 1) suggest that this rule may have been developed for adults and may not be useful in designing clothes for young children. Students can be asked to
FIGURE 1:
Changes in shape between infancy and adulthood, by age in years
create a rule that would work for young children. Because children's proportions change so rapidly with age, such a rule might include age as a variable.
There are numerous other rules of thumb: "The rule of 72" in finance, "Double the tax to get the tip" in a restaurant, "Magnetic north is true north" in navigation, and so on. Students can compare the results of these rules with actual data or investigate the accuracy and derivation of such rules in their areas of interest. For instance, The Joy of Cooking provides the following rule of thumb for cooking turkeys, "allow 20 to 25 minutes per pound for birds up to 6 pounds. For larger birds, allow 15 to 20 minutes per pound. For birds weighing over 16 pounds, allow 13 to 15 minutes per pound. In any case, add about 5 minutes to the pound if the bird you are cooking is stuffed'' (Rombauer & Becker, 1976). Students could explore the reasonableness of such predictions: might one conclude that a 5.9 pound bird requires (5.9) × (25) = 147.5 minutes, while a 6.1 pound bird requires no more than (6.1) × (20) = 122 minutes?
There are many other natural variations on the original problem as well. How sensitive is the car-length rule to what is assumed about the length of a car? Is the difference in average length of European versus U.S. sedans important to this rule of thumb? How should the two rules be modified for use on wet pavement? Questions might be raised about what happens if one car is traveling faster than the other or about the relationship between age and reaction time. In a state with a large number of retirees such as Florida, should the rules of thumb be the same as those in states with younger populations?

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Another issue concerns the usability of the two rules for following distance. If the two rules give essentially the same advice, is one easier to use in practice than the other? Is it easier to think in terms of distance measured in car lengths, picturing the space filled with cars, or to pick a marker such as a road sign or billboard, and count seconds? Opinions will vary as to which is the easier method.
References
Peitgen, H.-O. et al. (1992). Fractals for the classroom. New York: Springer-Verlag.
Rombauer, I.S. & Becker, M. R. (1976). The joy of cooking. New York: Bobbs-Merrill Company.