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 34
Overview
Students in grades 3 and 4 continue to develop the conceptual
building blocks of calculus primarily through their work with patterns
and changes over time. Students investigate a variety of patterns,
using physical materials and calculators as well as pictures. In some
of the patterns investigated, a constant is added to, or subtracted
from, each number to get the next number in the sequence. These
patterns, involving repeated operations, show linear growth;
when these patterns are represented with a bar graph, the tops of the
bars can be connected by a straight line. Examples of such patterns
include skip counting, starting the week with $5 and paying 75 cents
each day for lunch, or enumerating the multiples of 9. Other patterns
should involve multiplying or dividing a number by a constant to get
the next number in the sequence. These growing patterns illustrate
exponential growth, as is the pattern which results when you
start with two guppies (one male and one female) and the number of
guppies doubles each week. Patterns should also include looking at
changes over time, since these types of patterns are extremely
important not only in mathematics but also in science and social
studies. Students might chart the height of plants over time, the
number of teeth lost each month throughout the school year, or the
temperature outside the classroom over the course of several
months.
Students continue to develop their understanding of
measurement, gaining a greater understanding of the approximate
nature of measurement. Students can guess at the length of a stick
that is between 3 and 4 inches long, saying it is about 3 1/2 inches
long and recognizing when this is a better approximation than either 3
or 4. They can use grids of different sizes to approximate the area
of a puddle, recognizing that the smaller the grid the more accurate
the measurement. They can begin to consider how one might measure the
amount of water in a puddle, coming up with alternative strategies and
comparing them to see which would be more accurate. As they develop a
better understanding of volume, they may use cubes to build a solid,
build a second solid whose sides are all twice as long as the first,
and then compare the number of cubes used to build each solid. The
students may be surprised to find that it takes eight times as many
cubes to build the larger solid!
Students continue to develop their understanding of infinity
in grades 3 and 4. Additional work with counting sequences, skip
counting, and calculators further reinforces the notion that there is
always a bigger number. Taking half of something (like a rectangular
cake or a sheet of paper) repeatedly suggests that there is always a
number that is still closer to zero.
As students develop the conceptual underpinnings of calculus in
third and fourth grades, they are also working to develop their
understanding of numbers, patterns, measurement, data analysis, and
mathematical connections. Additional ideas for activities relating to
this standard can be found in the chapters discussing these other
standards.
Standard 15  Conceptual Building Blocks of Calculus  Grades 34
Indicators and Activities
The cumulative progress indicators for grade 4 appear below in
boldface type. Each indicator is followed by activities which
illustrate how it can be addressed in the classroom in grades 3 and
4.
Building upon knowledge and skills gained in the preceding grades,
experiences in grades 34 will be such that all students:
1. Investigate and describe patterns that
continue indefinitely.
 Using the constant multiplier feature of a calculator,
students see how many times 1 must be doubled before one million is
reached. They might first guess the number of steps to one million,
and to half a million.
 Students start with a long piece of string.
They fold it in half and cut it in two, setting aside one piece. Then
they take the remaining half, fold it in half, cut it apart, and set
aside half. They continue this process. They discuss how the length
of the string keeps getting smaller, half as much each time, so that
after about ten cuts, there is essentially no string left. Some
students may understand that the process could keep going for several
more steps, if we could only cut more carefully, and some may realize
that in theory the process could continue forever.
 Students count out 1, 2, 3, 4, 5, 6, ... , and recognize
that this pattern could continue forever. They also count out other
patterns, like the even numbers, or the square numbers, or
skipcounting by 3s starting with 2, and recognize that these patterns
also could be continued indefinitely.
 Students investigate the growth patterns of
sunflowers, pinecones, pineapples, or snails to study the natural
occurrence of spirals.
2. Investigate and describe how certain quantities
change over time.
 Students begin with a number like 5, add 3 to it, add 3 to it
again, and repeat this five times. They record the results in a table
and make a bar graph which represents the numbers that they have
generated. They draw a straight line connecting the tops of the bars.
They experiment with numbers other than 5 or 3 to see if the same
thing occurs.
 Students begin with a number less than 10, double it, and
repeat the doubling at least five times. They record the results of
each doubling in a table and make a bar graph which represents the
numbers they have generated. They discuss whether they can connect
the tops of the bars with a straight line.
 Students measure the temperature of a cup of water with ice
cubes in it every fifteen minutes over the course of a day. They
record their results (time passed and temperature) in a table and plot
this information on a coordinate grid to make a line graph. They
discuss how the temperature changes over time and why. Initially the
temperature will increase rapidly, but as the water warms up, its
temperature will increase more slowlyuntil it essentially reaches room
temperature.
 Students are given several examples of bar graphs with
straight lines connecting the tops of the bars. They are asked to
describe a motion scenario which reflects the data. For example, they
might indicate that a graph reflects their running to an afterschool
activity, staying there for an hour, and then slowly walking home to
do their chores.
 Students keep a monthly record of their height and record
the data collected on a bulletin board. At the end of the school
year, they describe what happened over time. They also find each
month the average height of all the students in the class, and discuss
how the change in average height over the year is similar to, and how
it is different from, the change in height over the year of the
individual students in the class.
 Students plant some seeds in vermiculite and some in soil.
They observe the plants as they grow, measuring their height each week
and recording their data in tables. They examine not only how the
height of each plant changes as time passes but also whether the seeds
in vermiculite or soil grow faster.
3. Experiment with approximating length, area, and
volume, using informal measurement instruments.
 Students measure the length of their classroom using their
paces and compare their results. They discuss what would happen if the
teacher measured the room with her pace.
 Students use pattern blocks to cover a drawing of a
dinosaur with as few blocks as they can. They record the number of
blocks of each type used in a table and then discuss their results,
making a frequency chart or bar graph of the total number of blocks
used by each pair of students. Then they try to cover the same
drawing with as many blocks as they can. They again record and
discuss their results and make a graph. They look for connections
between the numbers and types of blocks used each time. Some students
simply trade blocks (e.g., a hexagon is traded for six triangles),
while other students try to use all tan parallelograms since that
seems to be the smallest block. (It actually has the same area as the
triangle, however.)
 Students compare the volumes of a halfgallon milk carton,
a quart milk carton, a pint milk carton, and a halfpint milk carton.
They also measure the length of the side of the square base of each
carton and its height. They make a table of their results and look
for patterns. The students notice that the difference between the
height measurements is not the same as the difference between the
volume. The differences in volume grow more quickly than the
differences in the heights. They see how many small cubes or marbles
fill up each of their containers, and they try to explain why more
than twice as many fit into a quart container than a pint
container.
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.
