Instruction should be highly exploratory, based on numerical and geometric experiences that capitalize on both calculator and computer technology. Activities should be aimed at providing students with firm conceptual underpinnings of calculus rather that at developing manipulative techniques.

The development of calculus is one of the great intellectual achievements in history, especially with respect to its use in physics. Calculus is also increasingly being used in the social and biological sciences and in business. As students explore this area, they should develop an awareness of and appreciation for the historical origins and cultural contributions of calculus.

Students earlier study of patterns leads in high school to the study of finite and infinite processes. Students continue to look at linear growth patterns as they develop procedures for finding the sums of arithmetic series (e.g., the sum of the numbers from 1 to 100). They may consider this sum in many different ways, building different types of models. Some students may look at 1 + 2 + 3 + ... + 100 geometrically by putting together two "staircases" to form a rectangle that is 100 by 101. Other students may look at the sum arithmetically by adding 1 + 2 + 3 + ... + 100 to 100 + 99 + 98 + ... + 1 and getting 100 pairs of numbers that add up to 101. Still others may look at the sum by finding the limit of the sequence of partial sums. Students also look at exponential growth as they develop procedures for finding the sum of finite and infinite geometric series (e.g., 2 + 4 + 8 + 16 + 32 or 6 + 3 + 3/2 + ... or finding the total distance traveled by a bouncing ball). Students' work with patterns and infinity also includes elaborating on the intuitive notion of limit that has been addressed in the earlier grades.

High school students further develop their understanding of change over time through informal activities that focus on the understanding of interrelationships. Students need to collect data, generate graphs, and analyze the results for real-world situations that can be described by linear, quadratic, trigonometric, and exponential models. Some of the types of situations that should be analyzed include motion, epidemics, carbon dating, pendulums, and biological and economic growth. Students need to recognize the basic models (y = mx + b, y = ax^2 + bx + c, y=sin(x), and y = 2^x) and be able to relate geometric transformations to the equations of these models. Students need to develop a thorough understanding of the idea of slope; for example, they need to be able compare the steepness of two graphs at various points on the graph. They also need to be able explain what the slope means in terms of the real-world situation described by a graph. For example, what information does the slope give for a graph of the levels of medicine in the bloodstream over time? Students also extend their understanding of the behavior of functions to include the concept of the continuity of a function, considering features such as removable discontinuities (holes or jumps), asymptotes, and corners.

Students in high school apply their understanding of approximation techniques not only with respect to numbers in the context of using initial portions of nonrepeating, nonterminating decimals but also with respect to measurement situations. Students further develop their understanding of significant digits and the arithmetic of approximate values. They also use repeated approximations to find the areas of irregular figures, including experimenting with situations in which they need to find the area under a curve.

Looking at the conceptual underpinnings of calculus provides an opportunity for high school students to pull together their experiences with data analysis, patterns, algebra, measurement, number sense, and numerical operations. It also provides the opportunity to apply technology to real-world situations and to gain experience with mathematics as a dynamic human endeavor.

Building on the K-8 expectations, experiences in grades 9-12 will be such that all students:

L. develop and use models based on sequences and series.

• Students work in groups to collect data about a bouncing ball. They first decide how to measure the height of each bounce and then record their data in a table. They notice the pattern of the heights and make two graphs, one of height vs. bounce and the other of total distance traveled vs. bounce. They describe the general behavior of each graph and decide to have their graphing calculators compute regression lines. They try several different types of regressions until they find the one that fits best. In their report, they describe what they did, their results, and why they think that the type of function they used to describe each graph is reasonable.
• Students use M&Ms to model decay. They spill a package of M&Ms on a paper plate and remove those with the M showing, recording the number of M&Ms removed. They put the remaining M&Ms in a cup, shake, and repeat the process until all of the M&Ms are gone. They plot the trial number versus the number of M&Ms removed and note that the graph represents an exponential function. Some of the students try out different equations until they find one that they think fits pretty well.

• Students investigate a situation in which a contractor is fined $400 if he is one day late completing a project, $475 more if he is two days late, $550 more if he is three days late, and so on. They want to find out how much he will lose if he is two weeks late finishing the job. Some of the students solve the problem by first deciding that he will owe $400 + 14($75) = $1450 on the fourteenth day and then writing the following:

             400 +  475 +  550 + ... + 1300 + 1375 + 1450
            1450 + 1375 + 1300 + ... +  550 +  475 +  400
            ---------------------------------------------
            1850 + 1850 + 1850 + ... + 1850 + 1850 + 1850
  

  They decide that they have 14 pairs of numbers, each of which adds up to 1850. This gives them $25,900 which they divide in half (since they added up two sequences) to find the answer, $12,950. Another group decides that they will multiply 14 x 400, since he would owe the basic $400 each day, and then add on 75 (1 + 2 + 3 + ... + 13 + 14).

• Students investigate patterns in finding the sum 9 + 3 + 1 + 1/3 + 1/9 + 1/27. Most start by computing the sum (364/27 or about 13.815) on their calculators. One of the students notices that 364 is half of one less than 27^2. They make a conjecture that you might be able to find the sum by taking the reciprocal of the last term, squaring it, subtracting one, and then dividing by two. They decide to test this conjecture on another problem, 3 + 1/3 + 1/27, however, and it does not work. One group of students decides to try something like Gauss's approach to arithmetic series; they multiply the whole sequence by 3 (since that is the ratio between any two terms) and write:

              3 series = 27 + 9 + 3 +   1 + 1/3 +  1/9
            - 1 series =      9 + 3 + 1/3 + 1/9 + 1/27
            ------------------------------------------
              2 series = 27 - 1/27
  

  They make a conjecture that if you take the ratio, r, of any two terms and multiply that by the first term in the series and then subtract the last term in the series, you will have (r-1) times the sum. Each group shares its approach and its results with the class. The teacher develops the conjecture of the group that tried the Gauss-type approach into the standard formula,
  S(n) = a (1 - r^n)/(1 - r).

• After investigating how to find the sum of a finite geometric sequence, students begin looking at infinite geometric sequences. They think about what will happen if r is larger than one as n gets bigger and decide that the sum will keep increasing. Then they think about what will happen if r is less than one. As n gets bigger, r^n will get closer to zero, so the sum will get closer to a/(1 - r). They confirm their conjecture by checking out the partial sums of some sequences.

• Students look for the fractional equivalent of .9999... by first considering that 1/3 = .3333... They decide that .9999... is three times as large and so should be equal to 3/3, or 1. This does not seem quite right to them, however, so they decide to look at .999... as a sequence:
  .9, .09, .009, .0009, ...
  They compute the partial sums and see that, indeed, the series keeps getting closer to 1. They decide that 1 is the limit of the sequence.
• Students consider the sequence 1/2, 1/4, 1/8, ... in different contexts. First, they look at it as representing a situation in which someone eats half of a pizza, then half of what is left, then half of what is left, etc. They decide that, while there will always be some of pizza left, when you consider it all together, the sum of the sequence must be the whole pizza. Then they look at tearing a sheet of paper in half repeatedly and decide that the limit there must also be the whole sheet, or 1.

• Students use a graphing calculator, together with a light probe, to develop the relationship between brightness of a light and distance from it. They do this by collecting data with the probe on the brightness of a light bulb at increasing distances and analyzing the graph generated on the calculator.
• Students learn about the Richter Scale for measuring earthquakes, focussing on its representation as an exponential function.
• Students use recursive definitions of functions in both geometry and algebra. For example, they define n! recursively as n! = n (n-1)! They use recursion to generate fractals in studying geometry. They may use patterns such as spirolaterals, the Koch Snowflake, the Monkey's Tree curve, the Chaos Game, or the Sierpinski Triangle. They may use Logo or other computer programs to iterate patterns, or they may use the graphing calculator. In study algebra, students consider the equation y = .1x + .6, starting with an x value of .6, and find the resulting y-value. Using this y-value as the new x-value, they then calculate its corresponding y-value, and so on. (The resulting values are .6, .66, .666, .6666, etc.--an approximation to the decimal value of 2/3!) Students investigate using other starting values for the same function; the results are surprising! They use other equations to repeat the procedure. They graph the results and investigate the behavior of the resulting functions, using a calculator to reduce the computational burden.
• Students grow mold and collect data on the area of a pie plate covered by the mold. They make a graph showing the percent of increase in the area vs. the days, as in the vignette, "Breaking the Mold." The students graph their data and find an equation that fits the data to their satisfaction.

• Students work in groups to investigate what size square to cut from each corner of a rectangular piece of cardboard in order to make the largest possible open-top box. They make models, record the size of the square and the volume for each model, and plot the points on a graph. They note that the relationship seems to be a polynomial function and make a conjecture about the maximum volume, based on the graph. The students also generate a symbolic expression describing this situation and check to see if it matches their data by using a graphing calculator.
• Students look at the effects of changing the coefficients of a trigonometric equation on the graph. For example, how is the graph of y = 4 sin x different from that of y = sin x? How is y = .2 sin x different from y = sin x? How are y = sin x + 4, y = sin x - 4, y = sin (x - 4), and y = sin (x + 4) each different from y = sin x? Students use graphing calculators to look at the graphs and summarize their conjectures in writing.
• Students study the behavior of functions of the form y = ax^n. They investigate the effect of "a" on the curve and the characteristics of the graph when n is even or odd. They use the graphing calculator to assist them and write a sentence summarizing their discoveries.
• Students begin with the graph of y = 2^x. They shift the graph up one unit and try to find the equation of the resulting curve. They shift the original graph one unit to the right and try to find the equation of that curve. They reflect the original graph across the x-axis and try to find the equation of that curve. Finally, they reflect the original graph across the y-axis and try to find the equation of the resulting curve. They describe what they have learned in their journals.

• Students collect data about the height of a ball that is thrown in the air and make a scatterplot of their data. They note that the points lie on a quadratic function and use their graphing calculators to find the curve of best fit. Then they make some conjectures about the speed at which the ball is traveling. They think that the ball is slowing down as it rises, stopping at the maximum point, and speeding up again as it falls.
• Students take on the role of "forensic mathematicians," trying to determine how tall a person would be whose femur is 17 inches long. They measure their own femurs and their heights, entering this data into a graphing calculator or computer and creating a scatterplot. They note that the data are approximately linear, so they find the y-intercept and slope from the graph and generate an equation that they think will fit the data. They graph their equation and check its fit. They also use the built-in linear regression procedure to find the line of best fit and compare that equation to the one they generated.
• Students plot the data from a table that gives the amount of alcohol in the bloodstream after drinking two beers over time. Different groups use different techniques to generate an equation for the graph; after some discussion, the class decides which equation they think is best. The students consider the following questions: What information does the slope give for this situation? Would that be important to know? Why or why not?
• Students investigate the effect of changing the radius of a circle upon its circumference by measuring the radius and the circumference of circular objects. They graph the values they have generated, notice that it is close to a straight line, and use the slope to develop an equation that describes that relationship. Then they discuss the meaning of the slope in this situation.

• Students play the game described in the vignette, "On the Boardwalk," in which a quarter is thrown onto a grid. If the quarter does not touch a line, then you win. The students collect data and make a graph of their results (size of squares vs. number of wins out of 100 tosses). They note that, although the graph looks pretty much like a straight line, it actually has a bit of curve in it. No matter how large the square is, the most you could possibly win would be 100 games out of 100, so there is an asymptote at y=100.
• The school store sells pencils for 15 cents each, but it has some bulk pricing available if you need more pencils. Ten pencils sell for $1, and twenty-five pencils sell for $2. The students make a table showing the cost of different numbers of pencils and then generate a graph of number of pencils vs. cost. The students note that the graph has discontinuities at ten and twenty-five, since these are the jump points for pricing. They also note that if you need at least seven pencils, it is better to buy the package of ten and if you need 17 or more, you should get the package of 25.
• Students make a table, plot a graph (number of people vs. cost), and look for a function to describe a situation in which the Student Council is sponsoring a Valentine's Day dance and must pay $300 to the band, no matter how many people come. They also must pay $4 per person for refreshments, with a minimum of 50 people. The students note that the cost will be $500 for anywhere from 0-50 people and then increase at a rate of $4 per person. They decide that this is a function with a corner and needs to be defined in pieces:
  

            f(x) = 500 for x=<50
            f(x) = 500 + 4x for x>50
  

• Students investigate finding the area under the curve y = x^2 + 1 between -1 and 1. They approximate the area geometrically by dividing it into rectangles 0.5 units wide. They find the height of each rectangle that fits under the curve and use it to find the areas. Then they find the height of each rectangle that contains the curve and use these measurements to find the areas. They realize that this gives them a range of values for the area under the curve. They refine this approximation by using narrower rectangles, such as 0.1.
• After some experience with collecting data about balls thrown into the air, students are given a table of data about a model rocket and its height at different times. They plot the data, find an equation that fits the data, and use the trace functions on their graphing calculators to find the maximum height.