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