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 56
Overview
Students in grades 5 and 6 extend and clarify their understanding
of patterns, measurement, data analysis, number sense, and algebra as
they further develop the conceptual building blocks of calculus. Many
of the basic ideas of calculus can be examined in a very concrete and
intuitive way in the middle grades.
Students in grades 5 and 6 should begin to distinguish between
patterns involving linear growth (where a constant is added or
subtracted to each number to get the next one) and exponential
growth (where each term is multiplied or divided by the same
number each time to get the next number). Students should recognize
that linear growth patterns change at a constant rate. For example, a
plant may grow one inch every day. They should also begin to see that
if these patterns are graphed, then the graph looks like a straight
line. They may model this line by using a piece of spaghetti and use
their graph to make predictions and answer questions about points that
are not included in their data tables. In contrast, exponential
growth patterns change at an increasingly rapid rate; if you start
with one penny and double that amount each day, you receive more and
more pennies each day as time goes on. Students should note that the
graphs of these situations are not straight lines. At this grade
level, students should also begin to imagine processes that could in
theory continue forever even though they could not be carried out in
practice; for example, although in practice a cake can be repeatedly
divided in half only about ten times, nevertheless it is possible to
imagine continuing to divide it into smaller and smaller pieces.
Many of the examples used should come from other subject areas,
such as science and social studies. Students might look at such linear
relationships as profit as a function of selling price, but they
should also consider nonlinear relationships such as the amount of
rainfall over time. Students should look at functions which have
"holes" or jumps in their graphs. For example, if students
make a table of the parking fees paid for various amounts of time and
then plot the results, they will find that they cannot just connect
the points; instead there are jumps in the graph where the parking fee
goes up. A similar situation exists for graphs of the price of a
postage stamp or the minimum wage over the course of the years. Many
of the situations investigated by students should involve such
changes over time. Students might, for example, consider the
speed of a fly on a spinning disk; as the fly moves away from the
center of the disk, he spins faster and faster. Students might be
asked to write a short narrative about the fly on the disk and draw a
graph of the fly's speed over time that matches their story.
As students begin to explore the decimal equivalents for fractions,
they encounter nonterminating decimals for the first time. Students
should recognize that calculators often use approximations for
fractions such as .33 for 1/3. They should look for patterns
involving decimal representations of fractions, such as recognizing
which fractions have terminating decimal equivalents and which do not.
Students should take care to note that pi is a nonterminating,
nonrepeating decimal; it is not exactly equal to 22/7 or 3.14, but
these approximations are fairly close to the actual value of pi
and can usually be used for computational purposes. The examination
of decimals extends students' understanding of infinity to
very small numbers.
Students in grades 5 and 6 continue to develop a better
understanding of the approximate nature of measurement.
Students are able to measure objects with increasing degrees of
accuracy and begin to consider significant digits by looking at the
range of possible values that might result from computations with
approximate measures. For example, if the length of a rectangle to
the nearest centimeter is 10 cmand its width to the nearest centimeter
is 5 cm, then the area is about 50 square centimeters. However, the
rectangle might really be as small as 9.5 cm x 4.5 cm, in which case
the area would only be 42.75 square centimeters, or it might be almost
as large as 10.5 cm x 5.5 cm, with an area of 57.75 square
centimeters. Students should continue to explore how to determine the
surface area of irregular figures; they might, for example, be asked
to develop a strategy for finding the area of their hand or foot.
They should do similar activities involving volume, perhaps looking
for the volume of air in a car. Most of their work in this area in
fifth and sixth grade will involve using squares or cubes to
approximate these areas or volumes.
Standard 15  Conceptual Building Blocks of Calculus  Grades 56
Indicators and Activities
The cumulative progress indicators for grade 8 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 5 and
6.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 56 will be such that all students:
4. Recognize and express the difference between
linear and exponential growth.
 Students develop a table showing the sales tax paid on
different amounts of purchases, graph their results, note that the
graph is a straight line, and recognize that this situation represents
a constant rate of change, or linear growth.
 Students make a table showing how much money they
would have at the end of each of eight years if $100 was invested at
the beginning and the investment grew by 10% each year. They note that
the graph of their data is not a straight line; this graph represents
exponential growth.
 Students make a table showing the value of a car as it
depreciates over time. They note that the graph of their data is not
a straight line; this graph represents "exponential
decay."
 Students are presented stories which represent real life
occurrences of linear and exponential growth and decay over time, and
are asked to construct graphs which represent the situation and
indicate whether the change is linear, exponential, or neither.
5. Develop an understanding of infinite
sequences that arise in natural situations.
 Students make equilateral triangles of different
sizes out of small equilateral triangles and record the number of
small triangles used for each larger triangle. These numbers are
called triangular numbers. If the following triangular
pattern is continued indefinitely, then the number of 1s in the first
row, the number of 1s in the first two rows, the number of 1s in the
first three rows, etc. form the sequence of triangular numbers. The
triangular numbers also emerge from the handshake problem: If each
two people in a room shake hands exactly once, how many
handshakes take place altogether? If the answers are listed for
2, 3, 4, 5, 6, 7, ... people, the numbers are again the triangular
numbers 1, 3, 6, 10, 15, 21, ... .
1
1 1
1 1 1
1 1 1 1
1 1 1 1 1
1 1 1 1 1 1
 Students imagine cutting a sheet of paper into half,
cutting the two pieces into half, cutting the four pieces into half,
and continuing this over and over again, for about 25 times. Then they
imagine taking all of the little pieces of paper and stacking them on
top of one another. Finally, they estimate how tall that stack would
be.
 Students describe, analyze, and extend the
Fibonacci sequence (1, 1, 2, 3, 5, 8, ...). They research occurrences
of this sequence in nature, such as sunflower seeds, the fruit of the
pineapple, and the rabbit problem. They create their own
Fibonaccilike sequences, using different starting numbers.
6. Investigate, represent, and use nonterminating
decimals.
 Students use their calculators to find the decimal
equivalent for 2/3 by dividing 2 by 3. Some of the students get an
answer of 0.66667, while others get 0.6666667. They do the problem by
hand to try to understand what is happening. They decide that
different calculators round off the answer after different numbers of
decimal places. The teacher explains that the decimal for 2/3 can be
written exactly as .666... .
 Students have been looking for the number of different
squares that can be made on a 5 x 5 geoboard and have come up with
1x1, 2x2, 3x3, 4x4, and 5x5 squares. One student finds a different
square, however, whose area is 2 square units. The students wonder
how long the side of the square is. Since they know that the area is
the length of the side times itself, they try out different numbers,
multiplying 1.4 x 1.4 on their calculators to get 1.89 and
1.5 x 1.5 to get 2.25. They keep adding decimal places,
trying to get the exact answer of 2, but find that they cannot, no
matter how many places they try!
7. Represent, analyze, and predict relations
between quantities, especially quantities changing over
time.
 Students study which is the better way to cool down a soda,
adding lots of ice at the beginning or adding one cube at a time at
one minute intervals. Each student first makes a prediction and the
class summarizes the predictions. Then the class collects the data,
using probes and graphing calculators or computers and displays the
results in table and graph form on the overhead. The students compare
the graphs and write their conclusions in their math notebooks. They
discuss the reasons for their results in science class.
 Students make a graph that shows the price of mailing a
letter from 1850 through 1995. Some of the students begin by simply
plotting points and connecting them but soon realize that the price of
a stamp is constant for a period of time and then abruptly jumps up.
They decide that parts of this graph are like horizontal lines. The
teacher tells them that mathematicians call this a "step
function"; another name for this kind of graph is a piecewise
linear graph because the graph consists of linear pieces.
 Students review Mark's trip home from school on his
bike. Mark spent the first few minutes after school getting his books
and talking with friends, and left the school grounds about five
minutes after school was over. He raced with Ted to Ted's house
and stopped for ten minutes to talk about their math project. Then he
went straight home. The students draw a graph showing the distance
covered by Mark with respect to time. Then, with the teacher's
help, the class constructs a graph showing the speed at which Mark
traveled with respect to time. The students then write their own
stories and generate graphs of distance vs. time and graphs of speed
vs. time.
8. Approximate quantities with increasing degrees of
accuracy.
 Students find the volume of a cookie jar by first using
Multilink cubes (which are 2 cm on a side) and then by using
centimeter cubes. They realize that the second measurement is more
accurate than the first.
 Students measure the circumference and diameter of a paper
plate to the nearest inch and then divide the circumference by the
diameter. They repeat this process, using more accurate measures each
time (to the nearest halfinch, to the nearest quarterinch,
etc.). They see that the quotients get closer and closer to
pi.
 Using a ruler, students draw an irregularly shaped
pentagon on squaregrid paper, taking care to locate the vertices
of the pentagon at grid points. They estimate the area of the
pentagon by counting the number of squares completely inside the
pentagon and adding to it an estimate of the number of full square
that the partial squares inside the pentagon would add up to.
Then they divide the pentagon into triangles and rectangles and find
the area of the pentagon as a sum of the areas of the triangles and
rectangles. They compare the results and explain any
difference.
9. Understand and use the concept of significant
digits.
 Students measure the length and width of a rectangle in
centimeters and find its area. Then they measure its length and width
in millimeters and find the area. They note the difference between
these two results and discuss the reasons for such a difference. Some
of the students think that, since the original measurements were
correct to the nearest centimeter, then the result would be correct to
the nearest square centimeter, while the second measurements would be
correct to the nearest square millimeter. However, when they
experiment with different rectangles, for example, one whose
dimensions are 3.2 by 5.2 centimeters, they find that the area of 15
square centimeters is not correct to the nearest centimeter.
 Students find the area of a "blob" using a square
grid. First, they count the number of squares that fit entirely
within the blob (no parts hanging outside). They say that this is the
least that the area could be. Then they count the number of squares
that have any part of the blob in them. They say that this is the
most that the area could be. They note that the actual area is
somewhere between these two numbers.
10. Develop informal ways of approximating the surface
area and volume of familiar objects, and discuss whether the
approximations make sense.
 Students trace around their hand on graph paper and count
squares to find an approximate value of the area of their hand. They
use graph paper with smaller squares to find a better
approximation.
 Students work in groups to find the surface area of a
leaf. They describe the different methods they have used to
accomplish this task. Some groups are asked to go back and reexamine
their results. When the class is convinced that all of the results
are reasonably accurate, they consider how the surface area of the
leaf might be related to the growth of the tree and its needs for
carbon dioxide, sunshine, and water.
 Each group of students is given a mixing bowl and asked to
find its volume. One group decides to fill the bowl with centimeter
cubes, packing them as tightly as they can and then to add a
little. Another group decides to turn the bowl upside down and try to
build the same shape next to it by making layers of centimeter cubes.
Still another group decides to fill the hollow 1000centimeter cube
with water and empty it into the bowl as many times as they can to
fill it; they find that doing this three times almost fills the bowl
and add 24 centimeter cubes to bring the water level up to the top of
the bowl.
11. Express mathematically and explain the impact of
the change of an object's linear dimensions on its
surface area and volume.
 While learning about area, the students became curious
about how many square inches there are in a square foot. Some
students thought it would be 12, while others thought it might be
more. They explore this question using squareinch tiles to make a
square that is one foot on each side. They decide that there are 144
square inches in a square foot; they make the connection with
multiplication, noticing that 144 is 12 x 12 and that
there are 12 inches in a foot. They realize that the square numbers
have that name because they are the areas of squares whose sides are
the whole numbers.
 Having measured the length, width, and height of the
classroom in feet, the students now must find how many cubic yards of
air there are. Some of the students convert their measurements to
yards and then multiply to find the volume. Others multiply first,
but find that dividing by 3 does not give a reasonable answer. They
make a model using cubes that shows that there are 27 cubic feet in a
cubic yard and divide their answer by 27, getting the same result as
the other students.
OnLine Resources

http://dimacs.rutgers.edu/archive/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 gradespecific activities submitted by New Jersey
teachers, and to provide a forum to discuss the Mathematics
Standards.
