New Jersey Mathematics Curriculum Framework
© Copyright 1996 New Jersey Mathematics Coalition

STANDARD 15 - CONCEPTUAL BUILDING BLOCKS OF CALCULUS

All students will develop an understanding of the conceptual building blocks of calculus and will use them to model and analyze natural phenomena.

Standard 15 -Conceptual Building Blocks of Calculus - Grades 9-12

Overview

This standard does not advocate the formal study of calculus in high school for all students or even for all college-intending students. Rather, it calls for providing opportunities for all students to informally investigate the central ideas of calculus: limit, the rate of change, the area under a curve, and the slope of a tangent line. Considering these concepts will contribute to a deeper understanding of the function concept and its usefulness in representing and answering questions about real-world situations.

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 an understanding of the underlying concepts 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 is extended 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 should 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. They should use Calculator Based Labs (CBLs) in conjunction with graphing calculators to gather and analyze data. Students should recognize the equations of the basic models (y = mx + b, y = ax2 + bx + c, y=sin x, and y = 2x) and be able to relate geometric transformations to the equations of these models. Students should 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.

Standard 15 - Conceptual Underpinnings of Calculus - Grades 9-12

Indicators and Activities

The cumulative progress indicators for grade 12 appear below in boldface type. Each indicator is followed by activities which illustrate how it can be addressed in the classroom in grades 9, 10, 11, and 12.

Building upon knowledge and skills gained in the preceding grades, experiences in grades 9-12 will be such that all students:

12. 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 note that the distance traveled involves adding together the heights of each of the bounces, and so is represented by a series. They describe the general behavior of each graph and have their graphing calculators compute various regression lines. 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 exponential decay. They spill a package of M&Ms on a paper plate and remove those with the M showing, and record 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 exponential functions until they find one that they think fits the data pretty well.

13. Develop and apply procedures for finding the sum of finite arithmetic series and finite and infinite geometric series.

  • 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. They recognize that this is an arithmetic series where the first term is $400 and each term is obtained from the preceding one by adding $75. They draw upon several techniques they have learned to add up the terms of this series. One method that they have discussed involves reversing the order of the terms of the series and adding the two series. Some of the students thus solve the problem by writing the fourteen terms of the series and underneath writing the same fourteen terms backwards, a technique sometimes called Gauss' method because, according to legend, he discovered it as a child while walking to the back of his class to perform his punishment of adding together the first hundred numbers. They obtain the following format:

         400  +  475 +  550 + ... + 1300 + 1375
         1375 + 1300 + 1225 + ... +  475 +  400
         -------------------------------------------------------------
         1775 + 1775 + 1775 + ... + 1775 + 1775
    

    They recognize that they have 14 pairs of numbers, each of which adds up to 1775. This gives them a total of $24,850 which they divide in half (since they added together both sequences) to find the answer, $12,425. Another group decides that they can separate out the 14 charges of $400, for a total of 14 x 400 = $5600, and then deal with the remainders $0 + 75 + 150 + 225 + ... + 975, or $75(0 + 1 + 2 + 3 + ... + 13); this series they recognize as (13x14)/2, so the total fine is $5600 + $75x91 or $5600 + $6825, for a grand total of $12,425. Still another group of students uses a formula for the sum of a finite arithmetic series.

  • Students are asked to find a method similar to Gauss' method to find the sum of the series 9 + 3 + 1 + 1/3 + 1/9 + 1/27. The students notice that this series is not an arithmetic series since different amounts have to be added in order to get the next term. They discover, however, that each term is 1/3 of the previous term, and they write down 1/3 of the series and arrive at:

         9 + 3 + 1 + 1/3 + 1/9
             3 + 1 + 1/3 + 1/9 + 1/27
         ------------------------------------------------------
    

    They subtract to get 9 - 1/27 or 242/27 . Since they subtracted 1/3 of the series from itself, this total is 2/3 of the sum of the series, so the sum is 3x242/2x27 or 121/9. The teacher uses this technique to motivate the standard formula for the sum of a finite geometric series, where a is the first term of the series and r is the common multiple:

    Sn = 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 realize that the same technique they used for the finite geometric series works for the infinite one as well. Thus for example, if we added the first 100 terms of the series by the method above, the sum would be 9 - 1/397, which is very close to 9. Since this sum is again 2/3 of the sum of the original series, the actual total is 27/2. For those students who are likely to use a formula, the teacher generalizes this discussion and tells them that the sum gets closer to a/(1 - r) as the number of terms expand. They confirm this conclusion by checking out the partial sums of some sequences.

14. Develop an informal notion of limit.

  • After a class discussion of the repeating decimal .9999 ... , the students are asked to write in their journals an "explanation to the skeptic" on why .9999 ... is equal to 1. Among their explanations: There is no room between .9999 ... and 1; .9999 ... is 3 times .3333 ... which everyone agrees is 1/3 ; if you take 10 times .9999 ... and subtract .9999 ... , you get 9 and 9 times .9999 ... - so 1 must be .9999 ... ; if you sum a geometric series whose first time is .9 and whose common multiple is .1 you get a/(1-r) which amounts to .9/(1-.9), or 9. Given all these convincing reasons, the class decides that the limit of the sequence is 1.

  • 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 theoretically there will always be some of pizza left, in the end it would be all gone. However, in practice, by the end often stages or so the entire pizza would in effect have disappeared. Similarly, if a sheet of paper is repeatedly torn in half, then in theory some part is always left; however, in practice, after about ten tearings the paper will have disappeared.

15. Use linear, quadratic, trigonometric, and exponential models to explain growth and change in the natural world.

  • Students use a graphing calculator, together with a light probe, to examine 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 then analyzing the graph generated on the calculator to see what kind of graph it is. They use other CBL probes to investigate the kinds of functions used to model a variety of real-world situations.

  • Students learn about the Richter Scale for measuring earthquakes, focusing on its relation to logarithmic and exponential functions, and why this kind of scale is used.

  • 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 studying algebra, students consider the equation y = .1x + .6, start with an x-value of .6, and find the resulting yvalue. Using this yvalue as the new xvalue, they then calculate its corresponding yvalue, 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 and repeat the procedure. They graph the results and investigate the behavior of the resulting functions, using a calculator to reduce the computational burden.

  • Students work through the Breaking the Mold lesson described in the Introduction to this Framework. They 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. The students graph their data and find an equation that fits the data to their satisfaction.

16. Recognize fundamental mathematical models (such as polynomial, exponential, and trigonometric functions) and apply basic translations, reflections, and dilations to their graphs.

  • 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 = axn. 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 = 2x. 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.

17. Develop the concept of the slope of a curve, apply slopes to measure the steepness of curves, interpret the meaning of the slope of a curve for a given graph, and use the slope to discuss the information contained in the graph.

  • 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. (An instructional unit addressing this activity can be found in the Keys to Success in the Classroom chapter of this Framework.)

  • Students plot the data from a table that gives the amount of alcohol in the bloodstream at various intervals of time after a person drinks two glasses of beer. 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.

18. Develop an understanding of the concept of continuity of a function.

  • Students work through the On the Boardwalk lesson found in the Introduction to this Framework. A quarter is thrown onto a grid made up of squares, and you win if the quarter does not touch a line. A grid is drawn on the floor using masking tape, and a circular paper plate is thrown onto the grid several hundred times to simulate the game. The activity is repeated several times, varying each time the size of the squares in the grid. The students collect data and make a graph of their results (size of squares vs. number of wins out of 100 tosses). The graph looks like a straight line, suggesting that as the size of the squares increases without bound, so does the percentage of "hits". But, of course, the percentage of hits cannot exceed 100%, so the line is actually curved, with 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

19. Understand and apply approximation techniques to situations involving initial portions of infinite decimals and measurement.

  • Students investigate finding the area under the curve y = x2 + 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.

On-Line Resources

http://dimacs.rutgers.edu/nj_math_coalition/framework.html/

The Framework will be available at this site during Spring 1997. In time, we hope to post additional resources relating to this standard, such as grade-specific activities submitted by New Jersey teachers, and to provide a forum to discuss the Mathematics Standards.


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